SEMESTER TEST

Chapters 1-5 (continued) NAME

16. The graph ofy = g(x) shownhere is made of line segmqntsjoined end to end. Graph thefunctions derivative.

y 16.

Find all values ofx for which the function y -is differentiable.

5x - 2 17.

[4.7, 4.7] by [-3.1, 3.1]

18. Letf(x) = x2 - x - 41. Which of the following describesthe behavior of fat x = 4?(A) differentiable 03) comer(C) cusp (D) vertical tangent(E) discontinuity

18.

19. Find dy, where y - (x + 2)(x + 4)

20. dy d2yFind (a) d~x and (b) ~, where y = x4 - 5x3 + 9x - 15.

19.

z0.(b)

21. The monthly profit (in dollars) of a certain artist is givenby P(x) = - 15x3 + 150x2 + 300x - 400, where x is thenumber of pieces sold.(a) GraphP(x) andP(x).(b) What is the marginal profit when 6 pieces are sold?(c) What is the profit when the marginal profit is

greatest?(d) What is the maximum profit possible?

~ Assume x is an integer.)

21. (a)

~

3 tan x 22.22. Find if y -- 4 - cos x"

23. Find dy if y = sec (3x2). 23.dx24. Find 0% g) at x = 4 iff(u) = u3 - 5 and u = g(x) = 3~x. 24.

25. A curve is defined parametrically by x = sin 3t, y = cos 3t,25.0 < t

SEMESTER TEST

Chapters 1=5 (continued)

8. Let g(x) = 2 sec (3x - 70 + 1. Determine (a) the period,0a) the domain, and (e) ~he range, and (d) draw the graphof the function.

8. (a)(e)

Solve the equation sin x = 0.2 on the interva/0 --< x < 27r.

Which of the following statements are true about thefunctionfwhose graph is shown to the right?

I. lira f(x)=-ilI. f(1) = limf(x)

IlL lira f(x) = 1(A) I and II(D) I only

03) I and lII (C) II and III(E) I, II, and III

11. Determine x~olim \(sinx2x +.3x + 4)."

~ 5 -- x2, x ~ 21..2. Letf(x) = ~x + 3, x > 2"

Find the limit off(x) as(a) x-4-% (b) x--~2-, (e) x--~2+, and (d) x--~.

9.

10.

11,

12. (a)

2] "

1 ,~.-o~f(x)

-2

NAME

x2 + 3x - 413o Letf(x) = x2 _ 4x + 3"

Give a formulaforthe extended functionthat isi3.

continuous at x = 1.

46

14. An object is dropped from a 75-ft cliff. Its height in feetabove the beach after t sec is given by h(O = 75 - 16t~.(a) Find the average velocity during the interval from

t= 1tot=2.(b) Find the instantaneous velocity at t = 2.

I3 - x2, x --< 215. Letf(x) = [4x - 9, x > 2Determine whether the graph ofy = f(x) has a tangentat x = 2. If not, explain why not.

Calculus Assessment

14. (a)(b)

15.

Addison Wesley Longman, Inc.

SEMESTERTESTChapters ontinue NAME

26. U~e impficit differentiation to find ~ if cos xy = 2x2 - 3y. 26.

27. Find ~ ify = tan-1 (x - 4). 27.

28. Let y = x . Use logarithmic dtfferenttataon to find --.

(A) x-SinX(cosx- lnx) (B) xesx(C~X - lnxsinx)(C) x=l+cSXcosx (D) xCsx(C~X + lnxsinx)

xsinx)eosxlnx29. Find ~ ify = e2x - In x2.

30. Find the (global) extreme values and where they occur.

28.

29.

30, Min: at x =Max: at x =

31. For y = -lx4 - 2x3 - 1-x2 - 3 find the exact intervals on4 3 2

which the function is~ -(a) increasing,

(c) concave up,Then find any(el local extreme values, (f) inflection points.

(b) decreasing,(d) concave down.

32 A cars odometer (mileage cotmter) reads exactly59.024 miles at 8:00 A.M. and 59,094 miles at 10:00 A.M.Assuming the cars position and velocity functions aredifferentiable, what theorem can be used to show thatthe car was traveling at exactly 35 mph at some timebetween 8:00 A.M. and 10:00 A.M.?

I. Intermediate Value Theorem for DerivativeslI. Extreme Value Theorem

Ill. Mean Value Theorem(A)I only (B) /I tiny (C) III only(D/ I and III (El II and III

4~ Calculus Assessment

31, (a)(b)

(e) Min: at x =Max: atx =

32.

Addison Wesley Longman, Inc.

SEMESTER TESTChapters 1-5 (continued) NAME

33. Find all possible functions with the derivativef(x) = 4x2 - 3 + sinx.

34. Sketch a possible graph of a continuous function f thathas domain [-3, 31, where f(1) = -2 and the graph of

= f(x) is shown below.

[M, 4] by [-3, 3]

35. A piece of cardboard measures 22- by 35-in. Two equalsquares are removed from the comers of a 22-in. side asshown in the figure. Two equal rectangles are removedfrom the other comers so that the tabs can be folded toform a rectangular box with a lid.(a) Write a formula V(x) for the volume of the box.(b) Find the domain of V for the problem situation.(e) Find the maximum volume and the value ofx that

gives it.

33.

34.

36 Find the linearization L(x) off(x) = 3x4 - 5x3 at x = 2. 36.(A~ L(x) = 8x - 8 (B) L(x) = 32x - 56(C) L(x) = 36x (D) L(x) = 36x + 8(E) L(x) = 36x - 64

37. You are using Newtons method to solve In x - 3 = 0. 37.If your first guess is x1 = 15, what value will you calculatefor the next approximation

38. Suppose that the edge lengths x, y, and z of a closedrectangular box are changing at the following rates:~-~ = 4 ft/sec, ~t = 0 ft/sec, dz _ -3 ft/sec.dt dtFind the rates at which the boxs (a) volume,(b) surface area, and (c) diagonal lengths = ~/x2 + y2 + z2 are changing at the instant whenx= 10 ft, y = 8 ft, andz = 5ft.

[4, 4] by [-3, 3]

35. (a).(b) Domain:(c) Volume:

22 in. base lid

35 in.

38.(b)(c)

Addison Wesley Longman, Inc. Calculus Assessment 49

SEMESTER TEST

Chapters 1-5 (continued) NAME

39. The table below shows the velocity of a car in anamusement park ride during the first 24 seconds of theride. Use the left-endpoint values (LRAM) to estimatethe distance traveled, usiug 6 intervals of length 4.

Time(sec) 0 4 8 12 16 20 24Velocity(ft/sec) 0 4 7 7 11 16 18

Express the limit as a definite integral.

lim ~. (2c, + ~) Ax, where P is any partitiou of

41.

42.

[7, 151.

39.

40.

Suppose thatfand g are continuous and that f4_a f(x) dx = 5, 41. (a)f41g(x) dx = -2, and f~g(x) dx = 3. Find each integral. (b)

(e) f l (X) &Find the average value of thefunction g(x) = 2 + ~ 6on the interval [-3, 3] withoutintegrating, by appealing tothe region between the graph , .....and the x-axis.

~

= g(xi42.

f_2.6 5 cos x43. Use NINT to evaluate dx.1.7 ex q- x244. Evaluate each integral using Part 2 of the Fundamental

Theorem of Calculus.

ff -- ll6x3/4dx(a) (3x2 12 x + 4) dx (b)-5~r/6

(c~ / csc 0 cot 0 dO

Find d~y if y = (3t2 -- 50 dr.

(A) 3X2 -- 5X (B) 6x3 -- 10x2 (C) 12x3 - 10x(D) 6x5 - 10x3 (E) 3x6 - 5x4

46. Use the Trapezoidal Rule with n = 6 to approximate the,3

of [ex dxvalue~o

50 Calculus Assessmen~

43.

~4. (a)~ (b)

(e)

46.

Addison W~esley Longman,

Inc.

Semester Test Chapters 1-5 Form A4

5x 13

2. (a) ~

~b) All reals (e) (-% 2]3. (B) 4, ~ 19.525 years

(b) One possible answer:

6. OIm posaible answer:

(c) (-%-1JUt,)

~2. (a) -~ (b) 1(c) ~ (d) 1

13. g(x) = ~

(x - 1)a20. (a) 4x~ - 15xz + 9 ~) 1~~ - 30x

(b) $480.00 (e) $1711.11(d) $3920.00

22. 12 see~ x - 3 see x - 3 ~ x shl x(4 - cos x)2

23. 6x see (3x2) tan (3x2) 24. 8125. y = ~!~x- 2 26.4x + y sinxy

1 + (x - 4)2 28.29. 2e~x - _2

31. (a) [0, ~) (~) (--~, 01

(e) b/fin: -3 atx = 0Max: None

32. (D)33.f(x) = ~ -- 3x - cosx +C34. ~

35. (a) V(x) =

(b) Domain: (0, 11)

39. 180~V21

41, (a) 7 (b) 5(c) -5

42. 3~ + 2 43. ~ 7.498444. (a) -4 (b) 50~87

(e) - 145. (D) 46. ~ 19.482