trayectoria de la rana - wordpress.com diferencial. taller 2 1. para cada una de las funciones cuadr...

CALCULO DIFERENCIAL.TALLER 2

1. Para cada una de las funciones cuadraticas dadas encuentre el vertice, la grafica, el dominio yel recorrido.

49–50 ■ A quadratic function is given. (a) Use a graphing device tofind the maximum or minimum value of the quadratic function f,correct to two decimal places. (b) Find the exact maximum or mini-mum value of f, and compare it with your answer to part (a).

49.

50.

51–54 ■ Find all local maximum and minimum values of thefunction whose graph is shown.

51. 52.

53. 54.

55–62 ■ Find the local maximum and minimum values of the function and the value of x at which each occurs. State eachanswer correct to two decimal places.

55.

56.

57.

58.

59.

60.

61.

62.

A P P L I C A T I O N S63. Height of a Ball If a ball is thrown directly upward with a

velocity of 40 ft/s, its height (in feet) after t seconds is given by y � 40t � 16t2. What is the maximum height attained bythe ball?

V1x 2 �1

x2 � x � 1

V1x 2 �1 � x2

x3

U1x 2 � x2x � x2

U1x 2 � x16 � x

g1x 2 � x5 � 8x3 � 20x

g1x 2 � x4 � 2x3 � 11x2

f 1x 2 � 3 � x � x2 � x3

f 1x 2 � x3 � x

1

10 x

y

1

10

x

y

1

10 x

y

1

10 x

y

f 1x 2 � 1 � x � 12x2

f 1x 2 � x2 � 1.79x � 3.21

7. 8.

9–22 ■ A quadratic function is given. (a) Express the quad-ratic function in standard form. (b) Find its vertex and its x- and y-intercept(s). (c) Sketch its graph.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23–32 ■ A quadratic function is given. (a) Express the quadraticfunction in standard form. (b) Sketch its graph. (c) Find its maxi-mum or minimum value.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33–42 ■ Find the maximum or minimum value of the function.

33. 34.

35. 36.

37. 38.

39. 40.

41. 42.

43. Find a function whose graph is a parabola with vertex and that passes through the point .

44. Find a function whose graph is a parabola with vertex and that passes through the point .

45–48 ■ Find the domain and range of the function.

45. 46.

47. 48. f 1x 2 � �3x2 � 6x � 4f 1x 2 � 2x2 � 6x � 7

f 1x 2 � x2 � 2x � 3f 1x 2 � �x2 � 4x � 3

11, �8 213, 4 2

14, 16 211, �2 2

g1x 2 � 2x1x � 4 2 � 7f 1x 2 � 3 � x � 12 x2

f 1x 2 � �x2

3� 2x � 7h1x 2 � 1

2 x2 � 2x � 6

g1x 2 � 100x2 � 1500xf 1s 2 � s2 � 1.2s � 16

f 1t 2 � 10t2 � 40t � 113f 1t 2 � 100 � 49t � 7t2

f 1x 2 � 1 � 3x � x2f 1x 2 � x2 � x � 1

h1x 2 � 3 � 4x � 4x2h1x 2 � 1 � x � x2

g1x 2 � 2x2 � 8x � 11g1x 2 � 3x2 � 12x � 13

f 1x 2 � 1 � 6x � x2f 1x 2 � �x2 � 3x � 3

f 1x 2 � 5x2 � 30x � 4f 1x 2 � 3x2 � 6x � 1

f 1x 2 � x2 � 8x � 8f 1x 2 � x2 � 2x � 1

f 1x 2 � 6x2 � 12x � 5f 1x 2 � �4x2 � 16x � 3

f 1x 2 � 2x2 � x � 6f 1x 2 � 2x2 � 20x � 57

f 1x 2 � �3x2 � 6x � 2f 1x 2 � 2x2 � 4x � 3

f 1x 2 � �x2 � 4x � 4f 1x 2 � �x2 � 6x � 4

f 1x 2 � x2 � 2x � 2f 1x 2 � x2 � 4x � 3

f 1x 2 � �x2 � 10xf 1x 2 � 2x2 � 6x

f 1x 2 � x2 � 8xf 1x 2 � x2 � 6x

1

10 x

y

1

10 x

yf 1x 2 � 3x2 � 6x � 1f 1x 2 � 2x2 � 4x � 1

230 C H A P T E R 3 | Polynomial and Rational Functions

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How many trees should be planted per acre to obtain the maxi-mum yield of apples?

70. Agriculture At a certain vineyard it is found that eachgrape vine produces about 10 pounds of grapes in a seasonwhen about 700 vines are planted per acre. For each additionalvine that is planted, the production of each vine decreases byabout 1 percent. So the number of pounds of grapes producedper acre is modeled by

where n is the number of additional vines planted. Find thenumber of vines that should be planted to maximize grapeproduction.

71–74 ■ Use the formulas of this section to give an alternativesolution to the indicated problem in Focus on Modeling: Modelingwith Functions on pages 220–221.

71. Problem 21 72. Problem 22

73. Problem 25 74. Problem 24

75. Fencing a Horse Corral Carol has 2400 ft of fencing tofence in a rectangular horse corral.(a) Find a function that models the area of the corral in terms

of the width x of the corral.(b) Find the dimensions of the rectangle that maximize the

area of the corral.

76. Making a Rain Gutter A rain gutter is formed by bend-ing up the sides of a 30-inch-wide rectangular metal sheet asshown in the figure.(a) Find a function that models the cross-sectional area of the

gutter in terms of x.(b) Find the value of x that maximizes the cross-sectional area

of the gutter.(c) What is the maximum cross-sectional area for the gutter?

x

30 in.

x 1200 – x

A1n 2 � 1700 � n 2 110 � 0.01n 2

64. Path of a Ball A ball is thrown across a playing field froma height of 5 ft above the ground at an angle of 45º to the hori-zontal at a speed of 20 ft/s. It can be deduced from physicalprinciples that the path of the ball is modeled by the function

where x is the distance in feet that the ball has traveledhorizontally.(a) Find the maximum height attained by the ball.(b) Find the horizontal distance the ball has traveled when it

hits the ground.

65. Revenue A manufacturer finds that the revenue generatedby selling x units of a certain commodity is given by the func-tion , where the revenue is measuredin dollars. What is the maximum revenue, and how many unitsshould be manufactured to obtain this maximum?

66. Sales A soft-drink vendor at a popular beach analyzes hissales records and finds that if he sells x cans of soda pop inone day, his profit (in dollars) is given by

What is his maximum profit per day, and how many cans musthe sell for maximum profit?

67. Advertising The effectiveness of a television commercialdepends on how many times a viewer watches it. After someexperiments an advertising agency found that if the effective-ness E is measured on a scale of 0 to 10, then

where n is the number of times a viewer watches a given com-mercial. For a commercial to have maximum effectiveness,how many times should a viewer watch it?

68. Pharmaceuticals When a certain drug is taken orally,the concentration of the drug in the patient’s bloodstream after t minutes is given by , where 0 t 240 and the concentration is measured in mg/L. Whenis the maximum serum concentration reached, and what is thatmaximum concentration?

69. Agriculture The number of apples produced by each treein an apple orchard depends on how densely the trees areplanted. If n trees are planted on an acre of land, then eachtree produces 900 � 9n apples. So the number of apples produced per acre is

A1n 2 � n1900 � 9n 2

C1t 2 � 0.06t � 0.0002t2

E1n 2 � 23 n � 1

90 n2

P1x 2 � �0.001x2 � 3x � 1800

R1x 2R1x 2 � 80x � 0.4x2

x

5 ft

y � �

32

120 2 2 x 2 � x � 5

S E C T I O N 3 . 1 | Quadratic Functions and Models 231


Figura 1

Ejercicio 47

48 La bala de cañón humana En la década de 1940, la exhibi-

ción de la bala de cañón humana fue ejecutada regularmente

por Emmanuel Zacchini para el circo Ringling Brothers and

Barnum & Bailey. La punta del cañón se elevaba 15 pies del

suelo y la distancia horizontal total recorrida era de 175

pies. Cuando el cañón se apuntaba a un ángulo de 45°, una

ecuación del vuelo parabólico (vea la figura) tenía la forma

(a) Use la información dada para hallar una ecuación del

vuelo.

(b) Encuentre la altura máxima alcanzada por la bala de

cañón humana.

Ejercicio 48

49 Forma de un puente colgante Una sección de un puente

colgante tiene su peso uniformemente distribuido entre to-

rres gemelas que están a 400 pies entre sí y se elevan 90 pies

sobre la calzada horizontal (vea la figura). Un cable tendido

entre los remates de las torres tiene la forma de una parábola

y

x175�

y � ax2 � x � c.

y

x9

3

Trayectoria de la rana

y su punto central está 10 pies sobre la calzada. Suponga

que se introducen ejes de coordenadas, como se ve en la fi-

gura.

Ejercicio 49

(a) Encuentre una ecuación para la parábola.

(b) Nueve cables verticales igualmente espaciados se usan

para sostener el puente (vea la figura). Encuentre la lon-

gitud total de estos soportes.

50 Diseño de una carretera Unos ingenieros de tránsito están

diseñando un tramo de carretera que conectará una calzada

horizontal con una que tiene una pendiente del 20% es

decir, pendiente , como se ilustra en la figura. La transi-

ción suave debe tener lugar sobre una distancia horizontal

de 800 pies, con una pieza parabólica de carretera empleada

para conectar los puntos A y B. Si la ecuación del segmento

parabólico es de la forma , se puede de-

mostrar que la pendiente de la recta tangente en el punto

P(x, y) sobre la parábola está dada por .

(a) Encuentre una ecuación de la parábola que tiene una

recta tangente de pendiente 0 en A y en B.

(b) Encuentre las coordenadas de B.

Ejercicio 50

m � 0 AB

800�

m � Q

y

x

15

m � 2ax � b

y � ax 2 � bx � c

15�

�

y

x

90�

400�

226 C A P Í T U L O 3 F U N C I O N E S Y G R Á F I C A S

Swokowski_03C_3R.qxd 15/1/09 2:18 PM Page 226

Figura 2

2. Un agricultor desea poner una cerca alrededor de un campo rectangular, si el agricultor tiene2400 mts de cerca, (a) Encuentre una funcion que exprese el area de la region rectangularen funcion del ancho x (Figura 1.) (b) Encuentre las dimensiones del campo rectangular quemaximizan el area.

3. Los vuelos de animales saltarines tıpicamente tienen trayectorias parabolicas. La figura 2 ilustrael salto de una rana sobrepuesto en un plano de coordenadas. La longitud del salto es de 9 mtsy la maxima altura desde el suelo es 3 mts. Encuentre una ecuacion para la trayectoria de larana.

4. Encuentre todos los ceros racionales y factorice los siguientes polinomios:

a) p(x) = x3 + 3x2 − 4 b) p(x) = x3 − 7x2 + 14x− 8

c) p(x) = x4 − 5x2 + 4 d) p(x) = x4 − 2x3 − 3x2 + 8x− 4

e) p(x) = x4 + 6x3 + 7x2 − 6x− 8 f) p(x) = 2x3 + 7x2 + 4x− 4

5. Encuentre, sin usar calculadora, el valor de cada funcion trigonometrica.

S O L U T I O N Since tan t � sin t/cos t, we need to write sin t in terms of cos t. By thePythagorean identities we have

Solve for sin2 t

Take square roots

Since sin t is negative in Quadrant III, the negative sign applies here. Thus,

NOW TRY EXERCISE 55 ■

tan t �sin t

cos t�

�21 � cos2 t

cos t

sin t � �21 � cos2 t

sin2 t � 1 � cos2 t

sin2 t � cos2 t � 1

384 C H A P T E R 5 | Trigonometric Functions: Unit Circle Approach

C O N C E P T S1. Let be the terminal point on the unit circle determined

by t. Then , , and

.

2. If is on the unit circle, then .

So for all t we have .

S K I L L S3–4 ■ Find sin t and cos t for the values of t whose terminalpoints are shown on the unit circle in the figure. In Exercise 3,t increases in increments of p/4; in Exercise 4, t increases in increments of p/6. (See Exercises 21 and 22 in Section 5.1.)

3. 4.

5–24 ■ Find the exact value of the trigonometric function at thegiven real number.

5. (a) (b) (c)

6. (a) (b) (c)

7. (a) (b) (c)

8. (a) (b) (c) cos 7p

3cos a�

5p

3bcos

5p

3

sin 11p

6sin a�

p

6bsin

7p

6

tan 5p

6cos

5p

6sin

5p

6

tan 2p

3cos

2p

3sin

2p

3

y

x1_1

1

_1

π6t=

y

x1_1

1

_1

π4t=

sin2 t � cos2 t �

x 2 � y2 �P 1x, y 2tan t �

cos t �sin t �

P 1x, y 29. (a) (b) (c)

10. (a) (b) (c)

11. (a) (b) (c)

12. (a) (b) (c)

13. (a) (b) (c)

14. (a) (b) (c)

15. (a) (b) (c)

16. (a) (b) (c)

17. (a) (b) (c)

18. (a) (b) (c)

19. (a) (b) (c)

20. (a) (b) (c)

21. (a) (b) (c)

22. (a) (b) (c)

23. (a) (b) (c)

24. (a) (b) (c) cot 25p

2cos

25p

2sin

25p

2

tan 15pcos 14psin 13p

sec 4psec psec1�p 2csc

3p

2csc p

2csc a�

p

2b

tan 5p

4sec

5p

4sin

5p

4

cot a�

p

4bcsc a�

p

4bcos a�

p

4b

cot 5p

3cot

2p

3cot a�

p

3b

tan 11p

6tan

7p

6tan

5p

6

csc 7p

6sec

7p

6cos

7p

6

sec a�

p

3bcsc

11p

3sec

11p

3

cot a�

3p

2bcos a�

3p

2bsin a�

3p

2b

cot a�

p

2bcos a�

p

2bsin a�

p

2b

tan a�

p

3bsec a�

p

3bcos a�

p

3b

cot 7p

3csc

7p

3sin

7p

3

sin 7p

4sin

5p

4sin

3p

4

cos 7p

4cos

5p

4cos

3p

4

5 . 2 E X E R C I S E S


6. Encuentre la grafica de cada una de las funciones.

396 C H A P T E R 5 | Trigonometric Functions: Unit Circle Approach

C O N C E P T S1. The trigonometric functions and have

amplitude and period . Sketch a graph ofeach function on the interval .

2. The trigonometric function has amplitude

and period .

S K I L L S3–16 ■ Graph the function.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17–28 ■ Find the amplitude and period of the function, andsketch its graph.

17. y � cos 2x 18. y � �sin 2x

19. y � �3 sin 3x 20.

21. 22.

23. 24.

25. y � �2 sin 2px 26. y � �3 sin px

27. 28. y � �2 � cos 4px

29–42 ■ Find the amplitude, period, and phase shift of the func-tion, and graph one complete period.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38. y � 1 � cos a3x �p

2by �

1

2�

1

2 cos a2x �

p

3b

y � 2 sin a 2

3 x �

p

6by � 5 cos a3x �

p

4b

y � sin 1

2 a x �

p

4by � �4 sin 2 a x �

p

2b

y � 3 cos a x �p

4by � �2 sin a x �

p

6b

y � 2 sin a x �p

3by � cos a x �

p

2b

y � 1 � 12 cos px

y � 4 sin1�2x 2y � � 13 cos 13

x

y � 5 cos 14 xy � 10 sin 12

x

y � 12

cos 4x

h1x 2 � 0 sin x 0h1x 2 � 0 cos x 0g1x 2 � 4 � 2 sin xg1x 2 � 3 � 3 cos x

g1x 2 � � 23 cos xg1x 2 � �

12 sin x

g1x 2 � 2 sin xg1x 2 � 3 cos x

f 1x 2 � �1 � cos xf 1x 2 � �2 � sin x

f 1x 2 � 2 � cos xf 1x 2 � �sin x

f 1x 2 � 3 � sin xf 1x 2 � 1 � cos x

y � 3 sin 2x

�0, 2p�

y � cos xy � sin x

39. 40.

41. 42.

43–50 ■ The graph of one complete period of a sine or cosinecurve is given.

(a) Find the amplitude, period, and phase shift.

(b) Write an equation that represents the curve in the form

43. 44.

45. 46.

47. 48.

49. 50.

51–58 ■ Determine an appropriate viewing rectangle for eachfunction, and use it to draw the graph.

51. 52.

53. 54.

55. y � tan 25x 56. y � csc 40x

57. y � sin220x 58. y � 1tan 10px

f 1x 2 � cos1x/80 2f 1x 2 � sin1x/40 2f 1x 2 � 3 sin 120xf 1x 2 � cos 100x

y

x_ 14

14

34

5

_5

0x

y

_4

4

112_ 1

20

0

__

110

110

π4

π4

y

x

y

x

_ 12

12

2π3_ π

30

y

_3

3

2π 4π0 x0

_ 32

32

π6

π2

y

x

_2

2

π4 4

3π0

y

x

y

xπ 2π

_4

4

0

y � a sin k1x � b 2 or y � a cos k1x � b 2

y � cos ap2

� x by � sin1p � 3x 2y � 3 � 2 sin 31x � 1 2y � 3 cos p1x � 1

2 2

5 . 3 E X E R C I S E S

0

1

2π 0

1

2π


7. Simplifique la expresion trigonometrica.

E X A M P L E 7 Trigonometric Substitution

Substitute sin u for x in the expression and simplify. Assume that 0 � u� p/2.

S O L U T I O N Setting x � sin u, we have

Substitute x � sin u

Pythagorean identity

Take square root

The last equality is true because cos u � 0 for the values of u in question.

NOW TRY EXERCISE 91 ■

� cos u

� 2cos2 u

21 � x 2 � 21 � sin2

u

21 � x 2

498 C H A P T E R 7 | Analytic Trigonometry

C O N C E P T S1. An equation is called an identity if it is valid for

values of the variable. The equation is an alge-

braic identity, and the equation isa trigonometric identity.

2. For any x it is true that has the same value as .

We express this fact as the identity .

S K I L L S3–12 ■ Write the trigonometric expression in terms of sine andcosine, and then simplify.

3. cos t tan t 4. cos t csc t

5. sin u sec u 6. tan u csc u

7. tan2x � sec2x 8.

9. sin u � cot u cos u 10.

11. 12.

13–26 ■ Simplify the trigonometric expression.

13. 14. cos3x � sin2x cos x

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25.

26.cos x

sec x � tan x

tan u � cos1�u 2 � tan1�u 2

1 � cot A

csc A

2 � tan2 x

sec2 x

� 1

tan x cos x csc x 1 � sin u

cos u�

cos u

1 � sin u

sin x

csc x�

cos x

sec x

1 � csc x

cos x � cot x

sec x � cos x

tan x

sec2 x � 1

sec2 x

tan x

sec1�x 21 � cos y

1 � sec y

sin x sec x

tan x

cot u

csc u � sin u

sec u � cos u

sin u

cos2 u 11 � tan2

u 2sec x

csc x

cos xcos1�x 2sin2 x � cos2 x �

2x � x � x

27–28 ■ Consider the given equation. (a) Verify algebraically thatthe equation is an identity. (b) Confirm graphically that the equa-tion is an identity.

27. 28.

29–90 ■ Verify the identity.

29. 30.

31. 32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45. 46.

47.

48.

49. 11 � cos2 x 2 11 � cot2

x 2 � 1

sin4 u � cos4 u � sin2

u � cos2 u

1cot x � csc x 2 1cos x � 1 2 � �sin x

csc x � sin x � cos x cot x1

1 � sin2 y

� 1 � tan2 y

1 � sin x

1 � sin x� 1sec x � tan x 2 2

sec t � cos t

sec t� sin2

t

1sin x � cos x 2 4 � 11 � 2 sin x cos x 2 21sin x � cos x 2 2sin2

x � cos2 x

�sin2

x � cos2 x

1sin x � cos x 2 2

cos x

sec x�

sin x

csc x� 1

11 � cos b 2 11 � cos b 2 �1

csc2 b

1sin x � cos x 2 2 � 1 � 2 sin x cos x

tan u � cot u � sec u csc u

csc x 3csc x � sin1�x 2 4 � cot2 x

cot1�a 2 cos1�a 2 � sin1�a 2 � �csc a

cos1�x 2 � sin1�x 2 � cos x � sin x

sin B � cos B cot B � csc B

cot x sec x

csc x� 1

cos u sec u

tan u� cot u

tan x

sec x� sin x

sin u

tan u� cos u

tan y

csc y� sec y � cos y

cos x

sec x sin x� csc x � sin x

7 . 1 E X E R C I S E S


59. 60.

61–62 ■ The graph of is given. Match each equationwith its graph.

61. (a) (b)(c) (d)

62. (a) (b)(c) (d)

63. The graph of f is given. Sketch the graphs of the followingfunctions.(a) (b)(c) (d)(e) (f)

0

2

2 x

y

y � 12f 1x � 1 2y � f 1�x 2

y � �f 1x 2 � 3y � 2f 1x 2y � f 1x 2 � 2y � f 1x � 2 2

y

x3

3

_3

_3

_6 6

6 ➀

➁

➂

➃Ï

0

y � f 1�x 2y � f 1x � 4 2 � 3y � �f 1x � 4 2y � 1

3f 1x 2

y

x3

3

_3

_3

_6 6

6 ➀➁

➂

➃

Ï

0

y � �f 12x 2y � 2f 1x � 6 2y � f 1x 2 � 3y � f 1x � 4 2

y � f 1x 2

0 x

y

g

f(x)=x2 2

201

1 x

y

g

f(x)= x

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45–54 ■ A function f is given, and the indicated transformationsare applied to its graph (in the given order). Write the equation forthe final transformed graph.

45. ; shift upward 3 units

46. ; shift downward 1 unit

47. ; shift 2 units to the left

48. ; shift 1 unit to the right

49. ; shift 3 units to the right and shift upward 1 unit

50. ; shift 4 units to the left and shift downward 2 units

51. ; reflect in the y-axis and shift upward 1 unit

52. ; shift 2 units to the left and reflect in the x-axis

53. ; stretch vertically by a factor of 2, shift downward 2 units, and shift 3 units to the right

54. ; shrink vertically by a factor of , shift to the left 1 unit, and shift upward 3 units

55–60 ■ The graphs of f and g are given. Find a formula for thefunction g.

55. 56.

57. 58.

10 x

yg

f(x)=|x|2

110 x

yg

f(x)=|x|

x

y

1

10

g

f(x)=x3

x

y

gf(x)=x2 1

10

1 2

f 1x 2 � 0 x 0f 1x 2 � x2

f 1x 2 � x2

f 1x 2 � 24 x

f 1x 2 � 0 x 0f 1x 2 � 0 x 0f 1x 2 � 23 x

f 1x 2 � 1x

f 1x 2 � x3

f 1x 2 � x2

y � 3 � 21x � 1 2 2y � 1

2 1x � 4 � 3

y � 2 � 0 x 0y � 0 x � 2 0 � 2

y � 2 � 1x � 1

y � 3 � 12 1x � 1 2 2

y � 1x � 4 � 3

y � 1x � 3 2 2 � 5

y � 12 0 x 0

y � 3 0 x 0y � �52x

188 C H A P T E R 2 | Functions


Figura 3

59. 60.

61–62 ■ The graph of is given. Match each equationwith its graph.

61. (a) (b)(c) (d)

62. (a) (b)(c) (d)

63. The graph of f is given. Sketch the graphs of the followingfunctions.(a) (b)(c) (d)(e) (f)

0

2

2 x

y

y � 12f 1x � 1 2y � f 1�x 2

y � �f 1x 2 � 3y � 2f 1x 2y � f 1x 2 � 2y � f 1x � 2 2

y

x3

3

_3

_3

_6 6

6 ➀

➁

➂

➃Ï

0

y � f 1�x 2y � f 1x � 4 2 � 3y � �f 1x � 4 2y � 1

3f 1x 2

y

x3

3

_3

_3

_6 6

6 ➀➁

➂

➃

Ï

0

y � �f 12x 2y � 2f 1x � 6 2y � f 1x 2 � 3y � f 1x � 4 2

y � f 1x 2

0 x

y

g

f(x)=x2 2

201

1 x

y

g

f(x)= x

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45–54 ■ A function f is given, and the indicated transformationsare applied to its graph (in the given order). Write the equation forthe final transformed graph.

45. ; shift upward 3 units

46. ; shift downward 1 unit

47. ; shift 2 units to the left

48. ; shift 1 unit to the right

49. ; shift 3 units to the right and shift upward 1 unit

50. ; shift 4 units to the left and shift downward 2 units

51. ; reflect in the y-axis and shift upward 1 unit

52. ; shift 2 units to the left and reflect in the x-axis

53. ; stretch vertically by a factor of 2, shift downward 2 units, and shift 3 units to the right

54. ; shrink vertically by a factor of , shift to the left 1 unit, and shift upward 3 units

55–60 ■ The graphs of f and g are given. Find a formula for thefunction g.

55. 56.

57. 58.

10 x

yg

f(x)=|x|2

110 x

yg

f(x)=|x|

x

y

1

10

g

f(x)=x3

x

y

gf(x)=x2 1

10

1 2

f 1x 2 � 0 x 0f 1x 2 � x2

f 1x 2 � x2

f 1x 2 � 24 x

f 1x 2 � 0 x 0f 1x 2 � 0 x 0f 1x 2 � 23 x

f 1x 2 � 1x

f 1x 2 � x3

f 1x 2 � x2

y � 3 � 21x � 1 2 2y � 1

2 1x � 4 � 3

y � 2 � 0 x 0y � 0 x � 2 0 � 2

y � 2 � 1x � 1

y � 3 � 12 1x � 1 2 2

y � 1x � 4 � 3

y � 1x � 3 2 2 � 5

y � 12 0 x 0

y � 3 0 x 0y � �52x

188 C H A P T E R 2 | Functions


Figura 4

8. (Figuras 3 y 4) Dada la grafica de la funcion f(x), relacione cada funcion con su grafica.

9. Dada la grafica de g(x), encuentre la grafica de las siguientes funciones 72. Viewing rectangle 3�6, 64 by 3�4, 44(a) (b)

(c) (d)

73. If , graph the following functions in theviewing rectangle 3�5, 54 by 3�4, 44. How is each graph re-lated to the graph in part (a)?(a) (b) (c)

74. If , graph the following functions in theviewing rectangle 3�5, 54 by 3�4, 44. How is each graph re-lated to the graph in part (a)?(a) (b)(c) (d)(e)

75–82 ■ Determine whether the function f is even, odd, or nei-ther. If f is even or odd, use symmetry to sketch its graph.

75. 76.

77. 78.

79. 80.

81. 82.

83–84 ■ The graph of a function defined for x � 0 is given.Complete the graph for x � 0 to make (a) an even function and(b) an odd function.

83. 84.

85–86 ■ These exercises show how the graph of is obtained from the graph of .

85. The graphs of and are shown. Explain how the graph of g is obtained from the graph of f.

y

x2

4

_2

8

0_4

˝=|≈-4|

y

x2

4

_2

_4

8

0

Ï=≈-4

g 1x 2 � 0 x2 � 4 0f1x 2 � x2 � 4

y � f 1x 2y � 0 f 1x 2 0

x

y

110x

y

110

f 1x 2 � x �1x

f 1x 2 � 1 � 13 x

f 1x 2 � 3x3 � 2x2 � 1f 1x 2 � x3 � x

f 1x 2 � x4 � 4x2f 1x 2 � x2 � x

f 1x 2 � x3f 1x 2 � x4

y � f A� 12 xB

y � f 1�2x 2y � �f 1�x 2y � f 1�x 2y � f 1x 2

f 1x 2 � 22x � x2

y � f A12 xBy � f 12x 2y � f 1x 2

f 1x 2 � 22x � x2

y �1

2 1x � 3� 3y �

1

2 1x � 3

y �1

1x � 3y �

1

1x

64. The graph of g is given. Sketch the graphs of the followingfunctions.(a) (b)(c) (d)(e) (f)

65. The graph of g is given. Use it to graph each of the followingfunctions.(a) (b)

66. The graph of h is given. Use it to graph each of the followingfunctions.(a) (b)

67–68 ■ Use the graph of described on page 156 to graph the indicated function.

67. 68.

69–72 ■ Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graphin part (a)?

69. Viewing rectangle 3�8, 84 by 3�2, 84(a) (b)(c) (d)

70. Viewing rectangle 3�8, 84 by 3�6, 64(a) (b)(c) (d)

71. Viewing rectangle 3�4, 64 by 3�4, 44(a) (b)(c) (d) y � �

13 1x � 4 2 6y � �

13 x6

y � 13 x6y � x6

y � �3 0 x � 5 0y � �3 0 x 0y � � 0 x 0y � 0 x 0y � 4 � 214 x � 5y � 214 x � 5y � 14 x � 5y � 14 x

y � “ 14x‘y � “2x‘

f 1x 2 � “x‘

y

x

h

0 3_3

y � hA13 xBy � h13x 2

x

y

1

10

g

y � gA12 xBy � g12x 2

0

2

2 x

y

y � 2g1x 2y � �g1x 2y � g1x 2 � 2y � g1x � 2 2y � g1�x 2y � g1x � 1 2

S E C T I O N 2 . 5 | Transformations of Functions 189


10. Encuentre la grafica de f(x).

46

; ; Ejer. 47-52: Trace la gráfica de f.

47

48

49

50

51

52

Ejer. 53-54: El símbolo denota valores de la función deentero máximo. Trace la gráfica de f.

53 (a) (b)

(c) (d)

(e)

54 (a) (b)

(c) (d)

(e)

Ejer. 55-56: Explique por qué la gráfica de la ecuación no esla gráfica de una función.

55 56If , two different points If , two different pointshave x-coordinate x. have x-coordinate x.

x 0x 0x � � y x � y2

f �x� � ��x�

f �x� � �12 x�f �x� �

12 �x�

f �x� � �x� � 2f �x� � �x � 2�

f �x� � ��x�

f �x� � �2x�f �x� � 2�x�

f �x� � �x� � 3f �x� � �x � 3�

�x�

f �x� � �x � 3

�x2

�x � 4

si x � �2

si �2 x 1

si x 1

f �x� � �x � 2

x 3

�x � 3

si x � �1

si x 1

si x 1

f �x� � ��2x

x 2

�2

si x �1

si �1 � x 1

si x 1

f �x� � �3

�x � 1

�3

si x �2

si x � 2

si x 2

f �x� � ��1

�2

si x es un entero

si x no es un entero

f �x� � �3

�2

si x � �1

si x �1

y � �f �x � 4� � 2y � �f �x�y � f �x � 2� � 2

y

x

(a)(c)

(b) y � f (x)

Ejer. 57-58: Para la gráfica de mostrada en la fi-gura, trace la gráfica de .

57

58

Ejer. 59-62: Trace la gráfica de la ecuación.

59 60

61 62

63 Sea una función con dominio y

rango . Encuentre el dominio D y rango Rpara cada función. Suponga que y .

(a) (b), ,

(c) (d), ,

(e) (f ), ,

(g) (h), , R � �0, 8�D � ��2, 6�R � ��4, 8�D � ��6, 6�

y � f �x� y � f � x �R � ��8, 4�D � ��2, 6�R � ��4, 8�D � ��6, 2�

y � �f �x�y � f ��x�

R � ��7, 5�D � ��4, 4�R � ��3, 9�D � �1, 9�y � f �x � 2� � 3y � f �x � 3� � 1

R � ��4, 8�D � ��4, 12�R � ��16, 8�D � ��2, 6�y � f �1

2 x�y � �2f �x�

f �6� � �4f �2� � 8

R � ��4, 8�D � ��2, 6�y � f �x�

y � x � 1 y � 2x � 1

y � x3 � 1 y � 9 � x2

y

x

y

x

y � f (x) y � f (x)

3 . 5 G r á f i c a s d e f u n c i o n e s 211

Swokowski_03B_3R.qxd 16/1/09 3:33 PM Page 211

11. Cierto paıs grava los primeros $ 20,000 del ingreso de una persona a razon del 15 % y todo elingreso de mas de $ 20,000 se grava al 20 %. Encuentre una funcion T (x), definida a trozos, queespecifique el impuesto total sobre un ingreso de x pesos.

12. Encuentre.

Ejer. 1-2: Encuentre

(a) (b)

(c) (d)

1 , 15; ; 54;

2 ,

; ; ;

Ejer. 3-8: Encuentre (a) , , , y

(b) el dominio de , , y fg

(c) el dominio de

3 ,

4 ,

5 ,

6 ,

7 ,

8 ,

Ejer. 9-10: Encuentre(a) (b)

(c) (d)

9 ,; ; ; �x44x � 3�4x 2 � 4x � 1�2x 2 � 1

g�x� � �x 2f �x� � 2x � 1

(g g)(x)( f f )(x)

(g f )(x)( f g)(x)

g�x� �3x

x � 4f �x� �

x

x � 2

g�x� �x

x � 5f �x� �

2x

x � 4

g�x� � 2x � 4f �x� � 23 � 2x

g�x� � 2x � 5f �x� � 2x � 5

g�x� � x 2 � 3f �x� � x 2 � x

g�x� � 2x 2 � 1f �x� � x 2 � 2

f�g

f � gf � g

( f�g)(x)( fg)(x)( f � g)(x)( f � g)(x)

�95�45�14�4

g�x� � 2x � 1f �x� � �x 2

23�3g�x� � x 2f �x� � x � 3

( f�g)(3)( fg)(3)

( f � g)(3)( f � g)(3)

10 ,; ; ;


(c) (d)

11 ,; ; ; 10

12 ,; ; ; 101

13 ,; ; 304; 155

14 ,; ; 47; 256

15 ,; ; 31; 45

16 ,; ; 73; 186

17 ,; ; ; 3396

18 ,; ; ; 135

19 , 7; ; 7;

20 , 5; 25; 5; 25

Ejer. 21-34: Encuentre (a) y el dominio de y(b) y el dominio de .21 ,

, ; ,

22 ,

, ; , �15, ��x � 15 � 22x � 15��, �5� � �3, ��2x2 � 2x � 15

g�x� � x 2 � 2xf �x� � 2x � 15

��, 1� � �2, ��2x2 � 3x � 2��2, ��x � 2 � 32x � 2

g�x� � 2x � 2f �x� � x2 � 3xg f(g f )(x)

f g( f g)(x)

g�x� � x 2f �x� � 5

�7�7g�x� � �7f �x� � x �1443x 3 � 6x227x3 � 18x 2

g�x� � 3xf �x� � x 3 � 2x 2

�24128x3 � 20x8x3 � 20xg�x� � 2x 3 � 5xf �x� � 4x

75x 2 � 215x � 15615x2 � 5x � 3g�x� � 3x2 � x � 2f �x� � 5x � 7

4x2 � 6x � 98x 2 � 2x � 5g�x� � 2x � 1f �x� � 2x 2 � 3x � 4

36x2 � 24x � 412x 2 � 1g�x� � 4x 2f �x� � 3x � 1

15x 2 � 2075x 2 � 4g�x� � 5xf �x� � 3x 2 � 4

�6330x � 1130x � 3g�x� � 6x � 1f �x� � 5x � 2

�36x � 86x � 9g�x� � 3x � 7f �x� � 2x � 5

g( f (3))f (g(�2))

(g f )(x)( f g)(x)

x � 227x43x2 � 13x2 � 6x � 3g�x� � x � 1f �x� � 3x2


Hay dos ventajas de asignar las funciones en la forma citada líneas antes:

(1) No tuvimos en realidad que calcular la función polinomial a graficar, co-

mo hicimos en el ejemplo 7 de la sección 3.5.

(2) Con sólo cambiar los coeficientes en Y1 y Y3, fácilmente podemos exa-

minar su efecto sobre la gráfica de Y3.

Como ilustración del párrafo (2), el lector debe intentar graficar

cambiando Y1 a 3x, Y3 a y la pantalla a por y

luego graficando Y3, para obtener la figura 4.

��5, 1��1, 3�12 Y2y �

12 f �3x�

Figura 4por ��5, 1��1, 3�

3.7 E j e r c i c i o s

L


13. Encuentre.

Ejer. 1-2: Encuentre

(a) (b)

(c) (d)

1 , 15; ; 54;

2 ,

; ; ;

Ejer. 3-8: Encuentre (a) , , , y

(b) el dominio de , , y fg

(c) el dominio de

3 ,

4 ,

5 ,

6 ,

7 ,

8 ,


(c) (d)

9 ,; ; ; �x44x � 3�4x 2 � 4x � 1�2x 2 � 1

g�x� � �x 2f �x� � 2x � 1

(g g)(x)( f f )(x)

(g f )(x)( f g)(x)

g�x� �3x

x � 4f �x� �

x

x � 2

g�x� �x

x � 5f �x� �

2x

x � 4

g�x� � 2x � 4f �x� � 23 � 2x

g�x� � 2x � 5f �x� � 2x � 5

g�x� � x 2 � 3f �x� � x 2 � x

g�x� � 2x 2 � 1f �x� � x 2 � 2

f�g

f � gf � g

( f�g)(x)( fg)(x)( f � g)(x)( f � g)(x)

�95�45�14�4

g�x� � 2x � 1f �x� � �x 2

23�3g�x� � x 2f �x� � x � 3

( f�g)(3)( fg)(3)

( f � g)(3)( f � g)(3)

10 ,; ; ;


(c) (d)

11 ,; ; ; 10

12 ,; ; ; 101

13 ,; ; 304; 155

14 ,; ; 47; 256

15 ,; ; 31; 45

16 ,; ; 73; 186

17 ,; ; ; 3396

18 ,; ; ; 135

19 , 7; ; 7;

20 , 5; 25; 5; 25

Ejer. 21-34: Encuentre (a) y el dominio de y(b) y el dominio de .21 ,

, ; ,

22 ,

, ; , �15, ��x � 15 � 22x � 15��, �5� � �3, ��2x2 � 2x � 15

g�x� � x 2 � 2xf �x� � 2x � 15

��, 1� � �2, ��2x2 � 3x � 2��2, ��x � 2 � 32x � 2

g�x� � 2x � 2f �x� � x2 � 3xg f(g f )(x)

f g( f g)(x)

g�x� � x 2f �x� � 5

�7�7g�x� � �7f �x� � x �1443x 3 � 6x227x3 � 18x 2

g�x� � 3xf �x� � x 3 � 2x 2

�24128x3 � 20x8x3 � 20xg�x� � 2x 3 � 5xf �x� � 4x

75x 2 � 215x � 15615x2 � 5x � 3g�x� � 3x2 � x � 2f �x� � 5x � 7

4x2 � 6x � 98x 2 � 2x � 5g�x� � 2x � 1f �x� � 2x 2 � 3x � 4

36x2 � 24x � 412x 2 � 1g�x� � 4x 2f �x� � 3x � 1

15x 2 � 2075x 2 � 4g�x� � 5xf �x� � 3x 2 � 4

�6330x � 1130x � 3g�x� � 6x � 1f �x� � 5x � 2

�36x � 86x � 9g�x� � 3x � 7f �x� � 2x � 5

g( f (3))f (g(�2))

(g f )(x)( f g)(x)

x � 227x43x2 � 13x2 � 6x � 3g�x� � x � 1f �x� � 3x2


Hay dos ventajas de asignar las funciones en la forma citada líneas antes:

(1) No tuvimos en realidad que calcular la función polinomial a graficar, co-

mo hicimos en el ejemplo 7 de la sección 3.5.

(2) Con sólo cambiar los coeficientes en Y1 y Y3, fácilmente podemos exa-

minar su efecto sobre la gráfica de Y3.

Como ilustración del párrafo (2), el lector debe intentar graficar

cambiando Y1 a 3x, Y3 a y la pantalla a por y

luego graficando Y3, para obtener la figura 4.

��5, 1��1, 3�12 Y2y �

12 f �3x�

Figura 4por ��5, 1��1, 3�

3.7 E j e r c i c i o s

L


trayectoria de la rana - wordpress.com diferencial. taller 2 1. para cada una de las funciones cuadr...

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