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Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
15 Resolucioacuten de Desigualdades de primer grado con una incoacutegnita y de desigualdades cuadraacuteticas con una incoacutegnita
La solucioacuten de la desigualdad en una variable es el conjunto de todos los valores para los cuales la desigualdad es verdadera
Ejemplo
x + 3 gt 8x gt 8 ndash 3x gt 5
Unidad I Nuacutemeros RealesUnidad I Nuacutemeros Reales
3x + 3 gt 8 + 4x3x ndash 4x gt 8 ndash 3-x gt 5x lt -5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoFundamentos de Investigacioacuten Modulo ISeptiembre 2009
EjerciciosRepresenta en la recta numeacuterica y notacioacuten los siguientes conjuntos
1) x gt 13 2) -2 lt x le 3
3) x lt -3 u x gt 3 4) frac12 lt x lt 1
5) frac12 le x le 1
Instituto Tecnoloacutegico de Acapulco
Departamento Econoacutemico ndash AdministrativoFundamentos de Investigacioacuten Modulo ISeptiembre 2009
EjerciciosDeterminar su figura de las siguientes desigualdades
1) 1 ndash x lt 3 ndash 2x 2) x + 4 ge 4x - 2 -x + 2x lt 3 ndash 1 x ndash 4x ge -2 - 4 x lt 2 -3x ge -6
x le 2
3) 3x ndash 2 le 2 ndash x lt x + 6 4) 4 le 3x ndash 2 lt 133x + x le 2 + 2 -x ndash x lt 6 ndash 2 4 + 2 le 3x 3x lt 13 + 2 4x le 4 -2x lt 4 6 le 3x 3x lt 15 x le 1 x gt -2 -2 le x x lt 5 -2 lt x le 1 2 le x lt 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Fundamentos de Investigacioacuten Modulo ISeptiembre 2009
5) ndashx + 5 le 12 + 3x 2 44 (-x + 5) le 2 (12 + 3x)-4x + 20 le 24 + 6x -4x - 6x le 24 - 20-10x le 4x ge -4-10x ge - 25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad I
1) 3 gt x - 83 + 8 gt x 11 gt x
2) 3 (2 ndash x) gt 2 (3 + x) 6 ndash 3x gt 6 + 2x -3 x ndash 2x gt 6 ndash 6 -5x gt 0 x lt 0
3) 0 lt y ndash 4 lt 20 + 4 lt y lt 2 + 4
4 lt y lt 6
4) ysup2 gt 2ysup2 - 2 = 0ysup2 = 2y = plusmn radic2y lt - radic2 u y gt radic2
5) xsup2 ge 4xsup2 - 4 = 0x = plusmnradic4x le -2 u x ge 2
6) 4 lt xsup2 lt 9 radic4 lt x lt radic9
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoFundamentos de Investigacioacuten Modulo ISeptiembre 2009
EjerciciosRepresenta en la recta numeacuterica y notacioacuten los siguientes conjuntos
1) x gt 13 2) -2 lt x le 3
3) x lt -3 u x gt 3 4) frac12 lt x lt 1
5) frac12 le x le 1
Instituto Tecnoloacutegico de Acapulco
Departamento Econoacutemico ndash AdministrativoFundamentos de Investigacioacuten Modulo ISeptiembre 2009
EjerciciosDeterminar su figura de las siguientes desigualdades
1) 1 ndash x lt 3 ndash 2x 2) x + 4 ge 4x - 2 -x + 2x lt 3 ndash 1 x ndash 4x ge -2 - 4 x lt 2 -3x ge -6
x le 2
3) 3x ndash 2 le 2 ndash x lt x + 6 4) 4 le 3x ndash 2 lt 133x + x le 2 + 2 -x ndash x lt 6 ndash 2 4 + 2 le 3x 3x lt 13 + 2 4x le 4 -2x lt 4 6 le 3x 3x lt 15 x le 1 x gt -2 -2 le x x lt 5 -2 lt x le 1 2 le x lt 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Fundamentos de Investigacioacuten Modulo ISeptiembre 2009
5) ndashx + 5 le 12 + 3x 2 44 (-x + 5) le 2 (12 + 3x)-4x + 20 le 24 + 6x -4x - 6x le 24 - 20-10x le 4x ge -4-10x ge - 25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad I
1) 3 gt x - 83 + 8 gt x 11 gt x
2) 3 (2 ndash x) gt 2 (3 + x) 6 ndash 3x gt 6 + 2x -3 x ndash 2x gt 6 ndash 6 -5x gt 0 x lt 0
3) 0 lt y ndash 4 lt 20 + 4 lt y lt 2 + 4
4 lt y lt 6
4) ysup2 gt 2ysup2 - 2 = 0ysup2 = 2y = plusmn radic2y lt - radic2 u y gt radic2
5) xsup2 ge 4xsup2 - 4 = 0x = plusmnradic4x le -2 u x ge 2
6) 4 lt xsup2 lt 9 radic4 lt x lt radic9
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Departamento Econoacutemico ndash AdministrativoFundamentos de Investigacioacuten Modulo ISeptiembre 2009
EjerciciosDeterminar su figura de las siguientes desigualdades
1) 1 ndash x lt 3 ndash 2x 2) x + 4 ge 4x - 2 -x + 2x lt 3 ndash 1 x ndash 4x ge -2 - 4 x lt 2 -3x ge -6
x le 2
3) 3x ndash 2 le 2 ndash x lt x + 6 4) 4 le 3x ndash 2 lt 133x + x le 2 + 2 -x ndash x lt 6 ndash 2 4 + 2 le 3x 3x lt 13 + 2 4x le 4 -2x lt 4 6 le 3x 3x lt 15 x le 1 x gt -2 -2 le x x lt 5 -2 lt x le 1 2 le x lt 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Fundamentos de Investigacioacuten Modulo ISeptiembre 2009
5) ndashx + 5 le 12 + 3x 2 44 (-x + 5) le 2 (12 + 3x)-4x + 20 le 24 + 6x -4x - 6x le 24 - 20-10x le 4x ge -4-10x ge - 25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad I
1) 3 gt x - 83 + 8 gt x 11 gt x
2) 3 (2 ndash x) gt 2 (3 + x) 6 ndash 3x gt 6 + 2x -3 x ndash 2x gt 6 ndash 6 -5x gt 0 x lt 0
3) 0 lt y ndash 4 lt 20 + 4 lt y lt 2 + 4
4 lt y lt 6
4) ysup2 gt 2ysup2 - 2 = 0ysup2 = 2y = plusmn radic2y lt - radic2 u y gt radic2
5) xsup2 ge 4xsup2 - 4 = 0x = plusmnradic4x le -2 u x ge 2
6) 4 lt xsup2 lt 9 radic4 lt x lt radic9
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Fundamentos de Investigacioacuten Modulo ISeptiembre 2009
5) ndashx + 5 le 12 + 3x 2 44 (-x + 5) le 2 (12 + 3x)-4x + 20 le 24 + 6x -4x - 6x le 24 - 20-10x le 4x ge -4-10x ge - 25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad I
1) 3 gt x - 83 + 8 gt x 11 gt x
2) 3 (2 ndash x) gt 2 (3 + x) 6 ndash 3x gt 6 + 2x -3 x ndash 2x gt 6 ndash 6 -5x gt 0 x lt 0
3) 0 lt y ndash 4 lt 20 + 4 lt y lt 2 + 4
4 lt y lt 6
4) ysup2 gt 2ysup2 - 2 = 0ysup2 = 2y = plusmn radic2y lt - radic2 u y gt radic2
5) xsup2 ge 4xsup2 - 4 = 0x = plusmnradic4x le -2 u x ge 2
6) 4 lt xsup2 lt 9 radic4 lt x lt radic9
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad I
1) 3 gt x - 83 + 8 gt x 11 gt x
2) 3 (2 ndash x) gt 2 (3 + x) 6 ndash 3x gt 6 + 2x -3 x ndash 2x gt 6 ndash 6 -5x gt 0 x lt 0
3) 0 lt y ndash 4 lt 20 + 4 lt y lt 2 + 4
4 lt y lt 6
4) ysup2 gt 2ysup2 - 2 = 0ysup2 = 2y = plusmn radic2y lt - radic2 u y gt radic2
5) xsup2 ge 4xsup2 - 4 = 0x = plusmnradic4x le -2 u x ge 2
6) 4 lt xsup2 lt 9 radic4 lt x lt radic9
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
4 lt y lt 6
4) ysup2 gt 2ysup2 - 2 = 0ysup2 = 2y = plusmn radic2y lt - radic2 u y gt radic2
5) xsup2 ge 4xsup2 - 4 = 0x = plusmnradic4x le -2 u x ge 2
6) 4 lt xsup2 lt 9 radic4 lt x lt radic9
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
2 lt x lt 3 -3 lt x lt -2
7) 19 lt xsup2 lt frac14 radic19 lt x lt radic14 13 lt x lt 12
8) (x ndash 1)sup2 lt 4 x ndash 1 lt radic4 x ndash 1 lt - radic4 x lt 2 + 1 x lt -2 + 1 x lt 3 x lt -1
9) (x + 3) sup2 lt 2x + 3 lt radic2 x + 3 lt -radic2
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
x lt radic2 ndash 3 x lt -radic2 - 3
10) xsup2 - x lt 0 0 lt x lt 1
11) xsup2 - x ndash 2 ge 0 xsup2 - x ndash 2 = 0x= -(-1) plusmn radic(-1)sup2 - 4(-1)sup2(-2) 2 (1) x= 1 plusmn radic9 2 x= 1 plusmn 3 2 x= 1 + 3 = 2 2 x= 1 - 3 = -1 2
12) xsup2 - 5x + 6 le 0
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
(x ndash 3) (x ndash 2)X = 3X = 2
13) ndashx + 5 le 12 + 3x 2 44(-x + 5) le 2(12+3x)-4x + 20 le 24 + 6x-4x - 6x le 24 - 20-10 x le 410x ge -4X ge - ⅖
14) 6 ndash x lt 3x ndash 4 4 22(6 ndash x) lt 4(3x ndash 4)12 ndash 2x lt 12x ndash 16 -2x ndash 12x lt -16 ndash 12-14x lt -2814x gt 28X gt 2814X gt 2
15) |2x -3|= 7
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
2x ndash 3 = plusmn 72x = 7 + 3 2x = -7 + 3X = 102 x = -42X = 5 x = -2
16) |x|lt2-2 lt x lt 2
17) |x + 1 | ge 1 2X + 1 ge plusmn 1 21(x + 1) ge 2(1) 1(x + 1) ge 2(-1)X + 1 ge 2 X + 1 ge -2X ge 1 X le -3
18) |z + 2| lt 1Z + 2 lt plusmn 1
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Z lt 1 ndash 2 z lt -1 -2Z lt -1 z gt -3
19) |y + 3| ge frac12y + 3 ge plusmn frac12y ge frac12 - 3 y ge - frac12 - 3y ge -52 y le -72
20) |2z + 5| lt 42z + 5 lt plusmn 42z lt 4 ndash 5 2z lt -4 -5Z lt - frac12 z gt -92
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash Administrativo
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Calculo Diferencial Modulo ISeptiembre 2009
Ejercicios Unidad II
INSTRUCCIONES Hallar el Dominio y el Rango
1) f(x)= __1__ xsup2 - 1
xsup2 - 1 ne 0xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de Acapulco
x F(x) Sustitucioacuten -4 066 f(-4)= __1__ = __1__ = _1_
(-4)sup2 - 1 16 ndash 1 15-3 125 f(-3)= __1__ = __1__ = _1_
(-3)sup2 - 1 9 ndash 1 8-2 33 f(-2)= __1__ = __1__ = _1_
(-2)sup2 - 1 4 ndash 1 3-15 8 f(-15)= __1__ = __1__ = _1_
(-15)sup2 - 1 225 ndash 1 125-11 476 f(-11)= __1__ = __1__ = _1_
(-11)sup2 - 1 121 ndash 1 21-9 -526 f(-9)= __1__ = __1__ = _1_
(-9)sup2 - 1 81 ndash 1 -19-5 -133 f(-5)= __1__ = __1__ = _1_
(-5)sup2 - 1 25 ndash 1 -75-1 -11 f(-1)= __1__ = __1__ = _1_
(-1)sup2 - 1 01 ndash 1 -990 -1 f(0)= __1__ = __1__ = _1_
(0)sup2 - 1 0 ndash 1 -111 476 f(11)= __1__ = __1__ = _1_
(11)sup2 - 1 121 ndash 1 2115 8 f(15)= __1__ = __1__ = _1_
(15)sup2 - 1 225 ndash 1 12519 38 f(19)= __1__ = __1__ = _1_
(19)sup2 - 1 361 ndash 1 261 2 33 f(2)= __1__ = __1__ = _1_
(2)sup2 - 1 4 ndash 1 33 125 f(3)= __1__ = __1__ = _1_
(3)sup2 - 1 9 ndash 1 84 066 f(4)= __1__ = __1__ = _1_
(4)sup2 - 1 16 ndash 1 15
Unidad II FuncionesUnidad II Funciones
DOMINIOxx Є R ndash -1 y 1
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Departamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
2) F(x) = _x ndash 1_ xsup2 - 1xsup2 - 1 ne 0 x F(x) Sustitucioacuten
-4 -33 f(-4)= -4 ndash 1 = _-5_ = _-5_ (-4)sup2 - 1 16 ndash 1 15
-3 -5 f(-3)= -3 ndash 1 = _-4_ = _-4_ (-3)sup2 - 1 9 ndash 1 8
-2 -1 f(-2)= -2 ndash 1 = _-3_ = _-3_ (-2)sup2 - 1 4 ndash 1 3
-19 -11 f(-19)= -19 ndash 1 = _-29_ = _-29_ (-19)sup2 - 1 361 ndash 1 261
-15 -2 f(-15)= -15 ndash 1 = _-25_ = _-25_ (-15)sup2 - 1 225 ndash 1 125
-11 10 f(-11)= -11 ndash 1 = _-21_ = _-21_ (-11)sup2 - 1 121 ndash 1 21
-9 10 f(-9)= -9 ndash 1 = _-19_ = _-19 (-9)sup2 - 1 81 ndash 1 -19
-5 2 f(-5)= -5 ndash 1 = _-15_ = _-15_ (-5)sup2 - 1 25 ndash 1 -75
-1 11 f(-1)= -1 ndash 1 = _-11_ = _-11_ (-1)sup2 - 1 01 ndash 1 -99
0 1 f(0)= 0 ndash 1 = _-1_ = _-1_ (0)sup2 - 1 0 ndash 1 -1
11 47 f(11)= 11 ndash 1 = _1_ = _1_ (11)sup2 - 1 121 ndash 1 21
15 8 f(15)= -19 ndash 1 = _-29_ = _-29 (-15)sup2 - 1 225 ndash 1 125
2 3 f(2)= 2 ndash 1 = _1_ = _1_ (2)sup2 - 1 4 ndash 1 3
3 25 f(3)= 3 ndash 1 = _2_ = _2_ (3)sup2 - 1 9 ndash 1 8
4 2 f(4)= 4 ndash 1 = _3_ = _3_ (4)sup2 - 1 16 ndash 1 15
DOMINIOxx Є R ndash -1 y 1
RANGO
Unidad II FuncionesUnidad II Funciones
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
xsup2 ne 1xsup2 ne plusmn radic1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
3) F(x) = _x ndash 1_ xsup2 - 5x + 6
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
(x-2) (x-3) ne 0x1 ne 2x2 ne 3
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
4) F(x) = _xsup2 + 1_ xsup2 + x (x+x) (x+1) ne 0
x F(x) Sustitucioacuten8 23 f(8)=____ 8 ndash 1____ = ___7___ = _7_
(8)sup2 - 5(8) + 6 64 - 40 + 6 307 3 f(7)=____ 7 ndash 1____ = ___6___ = _6_
(7)sup2 - 5(7) + 6 49 - 35 + 6 206 41 f(6)=____ 6 ndash 1____ = ___5___ = _5_
(6)sup2 - 5(6) + 6 36 - 30 + 6 125 6 f(5)=____ 5 ndash 1____ = ___4___ = _4_
(5)sup2 - 5(5) + 6 25 ndash 25 + 6 64 15 f(4)=____ 4 ndash 1____ = ___3___ = _3_
(4)sup2 - 5(4) + 6 16 ndash 20 + 6 21 0 f(1)=____ 1 ndash 1____ = ___0___ = _0_
(1)sup2 - 5(1) + 6 1 ndash 5 + 6 20 -16 f(0)=____ 0 ndash 1____ = ___-1___ = _-1_
(0)sup2 - 5(0) + 6 0 ndash 0 + 6 6
DOMINIOxx Є R ndash 2 y 3
RANGO
Unidad II FuncionesUnidad II Funciones
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
x1 ne -1x2 ne 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
5) f(x) = _3x - 1_ 2x + 5
x F(x) Sustitucioacuten8 109 f(8)=_3(8) ndash 1 = _24 ndash 1_ = _23_
2(8) + 5 16 + 5 216 1 f(6)=_3(6) ndash 1 = _18 ndash 1_ = _17_
2(6) + 5 12 + 5 174 9 f(4)=_3(4) ndash 1 = _12 ndash 1_ = _11_
2(4) + 5 8 + 5 122 5 f(2)=_3(2) ndash 1 = _6 ndash 1_ = _5_
2(2) + 5 4 + 5 90 -2 f(0)=_3(0) ndash 1 = _0 ndash 1_ = _-1_
2(0) + 5 0 + 5 5-2 -7 f(-2)=_3(-2) ndash 1 = _-6 ndash 1_ = _-7_
2(-2) + 5 -4 + 5 1-4 43 f(-4)=_3(-4) ndash 1 = _-12 ndash 1_ = _-13_
2(-4) + 5 -8 + 5 -3-6 27 f(-6)=_3(-6) ndash 1 = _-18 ndash 1_ = _-19_
2(-6) + 5 -12 + 5 -7-8 22 f(-8)=_3(-8) ndash 1 = _-24 ndash 1_ = _-25_
2(-8) + 5 -16 + 5 -11
DOMINIOxx Є R ndash -25
RANGO
Unidad II FuncionesUnidad II Funciones
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
2x + 5 ne 02x ne -5X ne -⁵∕₂ -25
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
INSTRUCCIONES Hallar el Dominio y el Rango
1) P(x) = radicxx ge 0
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
2) P(x) = radicx - 1x - 1 ge 0x ge 1
x F(x) Sustitucioacuten0 0 f(0)= radic01 1 f(1)= radic12 141 f(2)= radic23 173 f(3)= radic34 2 f(4)= radic4
x F(x) Sustitucioacuten1 0 f(1)= radic1 ndash 1 = radic02 1 f(2)= radic2 ndash 1 = radic13 141 f(3)= radic3 ndash 1 = radic24 173 f(4)= radic4 ndash 1 = radic3
DOMINIOxx Є R ge 0
[0 infin)
RANGO 0 a 2
DOMINIOxx Є R ge 1
[1 infin)
RANGO 0 a 173
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3) P(x) = radicxsup2 - 1xsup2 - 1 ge 0xsup2 ge 1plusmnradicx ge 1 x₁ ge 1-x₂ ge 1
x F(x) Sustitucioacuten-8 79 f(-8) = radic(-8)sup2 - 1 = radic63-7 69 f(-7) = radic(-7)sup2 - 1 = radic48-6 59 f(-6) = radic(-6)sup2 - 1 = radic35-5 48 f(-5) = radic(-5)sup2 - 1 = radic24-4 38 f(-4) = radic(-4)sup2 - 1 = radic15-3 28 f(-3) = radic(-3)sup2 - 1 = radic8-2 17 f(-2) = radic(-2)sup2 - 1 = radic32 17 f(2) = radic(2)sup2 - 1 = radic33 28 f(3) = radic(3)sup2 - 1 = radic84 38 f(4) = radic(4)sup2 - 1 = radic155 48 f(5) = radic(5)sup2 - 1 = radic246 59 f(6) = radic(6)sup2 - 1 = radic357 69 f(7) = radic(7)sup2 - 1 = radic488 79 f(8) = radic(8)sup2 - 1 = radic63
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 173
RANGO 0 a 173
RANGO
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
x₂ le -1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
4) P(x) = radic1 - xsup21 - xsup2 ge 0-xsup2 ge -1xsup2 le 1plusmnradicx le 1 x₁ le 1
x F(x) Sustitucioacuten-1 0 f(-1) = radic1 - (-2)sup2 = radic00 1 f(0) = radic1 - (0)sup2 = radic11 0 f(1) = radic1- (1)sup2 = radic0
DOMINIOxx Є R -1 le x le 1
RANGO 0 a 1
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
-x₂ le 1x₂ ge -1
5) P(x) = radic1 - 9xsup21 - 9xsup2 ge 0-9xsup2 ge -1xsup2 le 19plusmnradicx le 19 x₁ le 19-x₂ le 19x₂ ge -19
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
6) P(x) = radic4x + 34x + 3 ge 04x ge -3x ge -34
x F(x) Sustitucioacuten- sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
0 1 f(0) = radic1 - 9(0)sup2 = radic1sup1₉ 094 f(- sup1₉) = radic1 - 9(- sup1₉)sup2 = radic⁸₉
x F(x) Sustitucioacuten-75 0 f(-75) = radic4(-75) + 3 = radic0-5 1 f(-5) = radic4(-5) + 3 = radic10 173 f(0) = radic4(0) + 3 = radic31 26 f(0) = radic4(1) + 3 = radic72 33 f(0) = radic4(2) + 3 = radic113 38 f(0) = radic4(3) + 3 = radic15
DOMINIOxx Є R -19 le x le 19
RANGO 094 a 1
DOMINIOxx Є R ge -34
RANGO 0 a infin
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
7) P(x) = radicx - 5x - 5 ge 0x ge 5
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
8) P(x) = sup3 radicx + 2x + 2 ge 0x ge -2
x F(x) Sustitucioacuten5 0 f(5) = radic5 ndash 5 = radic0
10 22 f(10) = radic10 ndash 5 = radic515 31 f(15) = radic15 ndash 5 = radic10
20 38 f(20) = radic20 ndash 5 = radic15
x F(x) Sustitucioacuten-2 0 f(-2) = sup3radic-2 + 2 = sup3 radic0-1 1 f(-1) = sup3radic-1 + 2 = sup3 radic10 125 f(0) = sup3radic0 + 2 = sup3 radic21 14 f(1) = sup3radic1 + 2 = sup3 radic32 15 f(2) = sup3radic2 + 2 = sup3 radic45 19 f(5) = sup3 radic5 + 2 = sup3 radic7
10 22 f(10) = sup3 radic10 + 2 = sup3 radic1215 25 f(15) = sup3 radic15 + 2 = sup3 radic1720 28 f(20) = sup3 radic20 + 2 = sup3 radic22
DOMINIOxx Є R ge 5
RANGO 0 a infin
DOMINIOxx Є R ge -2
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
9) P(x) = frac12 radicx + 4| -3x + 4 ge 0x ge - 4 x F(x) Sustitucioacuten
-4 -3 P(-4) = frac12 radic-4 + 4| -3 = frac12 radic0 -3-2 -229 P(-2) = frac12 radic-2 + 4| -3 = frac12 radic2 -30 -2 P(0) = frac12 radic0 + 4| -3 = frac12 radic4 -32 -17 P(2) = frac12 radic2 + 4| -3 = frac12 radic6 -34 -15 P(4) = frac12 radic4 + 4| -3 = frac12 radic8 -36 -14 P(6) = frac12 radic6 + 4| -3 = frac12 radic10 -38 -12 P(8) = frac12 radic8 + 4| -3 = frac12 radic12 -3
DOMINIOxx Є R ge -4
RANGO 0 a infin
RANGO 0 a infin
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
Ejercicios Unidad II
Unidad II FuncionesUnidad II Funciones
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
ldquoFunciones Trigonometricasrdquo
1- f(x) = 2senx
2- f(x) = senx + 1
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
3- f(x) = -senx
x F(x)0 0
45˚ = π4 141490˚ = π2 2
135˚ = 3π4 1414180˚ = π 0225˚ = π -1414
270˚ = 3π2 -2315˚ = 7π4 -1414360˚ = 2π 0
x F(x)0 1
45˚ = π4 17090˚ = π2 2
135˚ = 3π4 170180˚ = π 1225˚ = π 0292
270˚ = 3π2 0315˚ = 7π4 0292360˚ = 2π 1
x F(x)0 0
45˚ = π4 -0790˚ = π2 -1
135˚ = 3π4 -07180˚ = π 0225˚ = π 07
270˚ = 3π2 1315˚ = 7π4 07360˚ = 2π 0
RANGO -2 a 2
RANGO 0 a 2
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
4- f(x) = cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
5- f(x) = -cos x
x F(x)0 1
45˚ = π4 70790˚ = π2 0
135˚ = 3π4 -0707180˚ = π -1225˚ = π -0707
270˚ = 3π2 0315˚ = 7π4 0707360˚ = 2π 1
x F(x)0 -1
45˚ = π4 -0790˚ = π2 0
135˚ = 3π4 07180˚ = π 1225˚ = π 07
270˚ = 3π2 0315˚ = 7π4 -07360˚ = 2π -1
RANGO -1 a 1
RANGO -1 a 1
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
6- f(x) = 2 cos x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo IOctubre 2009
7- f(x) = cos x + 3
x F(x)0 2
45˚ = π4 14190˚ = π2 0
135˚ = 3π4 -141180˚ = π -2225˚ = π -141
270˚ = 3π2 0315˚ = 7π4 141360˚ = 2π 2
x F(x)0 4
45˚ = π4 3790˚ = π2 3
135˚ = 3π4 229180˚ = π 2225˚ = π 229
270˚ = 3π2 3315˚ = 7π4 37360˚ = 2π 4
RANGO -1 a 1
RANGO -1 a 1
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
8- f(x) = tan x
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Instrucciones1) Grafique2) Analice que pasa con el dibujo de la funcioacuten cuando el paraacutemetro a b c cambian
x F(x)0 2
45˚ = π4 1135˚ = 3π4 -1
180˚ = π 0225˚ = π 1
315˚ = 7π4 -1360˚ = 2π 0
RANGO 2 a 4
RANGO -1 a 2
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
1- f(x)= xsup2
Instituto Tecnoloacutegico de AcapulcoDepartamento Econoacutemico ndash AdministrativoCalculo Diferencial Modulo ISeptiembre 2009
Ejercicios
x F(x)-3 9-2 4-1 10 01 12 43 9
Unidad III LiacutemitesUnidad III Liacutemites
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
Aplique las propiedades de los liacutemites y resuelva los siguientes ejercicios
1) Lim 3x -3
2) Lim xx radic2
3) Lim xsup3x 2
4) Lim xsup2 + xx -3
5) Lim xsup3 + 3 x 1 x
6) Lim (xsup3 + 3) ( x - 1)x 1
7) Lim radicx + 2x 2
Lim 3 = 3 Lim c = cx -3 x a
Lim x = radic2 = 14142 Lim x = ax radic2 x a
Lim xsup3 = (lim x)sup3 = (lim 2)sup3= (2)sup3 = 8 Lim xⁿ = aⁿx 2 x a
Lim xsup2 + x = (lim x)sup2 + lim x = (-3)sup2 + (-3) = 6 Lim [ f(x) + g(x)]x radic2 x a
Lim xsup3 + 3 = (1)sup3 + 3 = 4 = 4 Lim f(x) = lim f(x) x 1 x 1 1 x a x a
lim g(x)x a
Lim xsup3 + 3 = (1)sup3 + 3) (1 ndash 1) = 0 Lim f(x) = lim f(x) lim g(x)x 1 x a x a
Lim radicx + 2 = radic2 + 2 = 2 Lim ⁿradicf(x) = ax 2 x a
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