Calculus AB-Exam 1 - Ms. Orloff

Time: 60 minutes Number of questions: 30

Calculus AB-Exam 1 Section I, Part A

NO CAlCULATOR MAY BE USF.D IN THIS PART OF THE EXAMINATION.

Directions: Sol~e each of the following problems. After examining the form of the choices, decide which is the best of the choices given.

In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of aU real numbers x for which f(x) is a real number.

1. Let fbe the function defined below, wbere a and b are constants. Iff is differentiable at x = 3, what is the value of a - b? '

(A) 6

{ax - b for x < 3

f(x) = x 2 -I bx for x ~ 3

(B) 9 (C) 15

y

(D) 24

2. The graph of a piecewise-linear function f, for - 3 ::s; x ::s; 4, is shown above. If g(x) := j~1 f( t) dt, which of the following values is the least?

3.

(A) g( -3)

3

1 ). dx .--x 3

2

(A) 5 72

(B) g(O)

(B) 5 36

(C) gO) (D) g(4)

5 (D) 72

--------------------Copyright © 2017 Pearson Fducation, Inc.

221

4. J is continuous for a :5; x :::: b but not differentiable for some c such that a < c < b. Which of the following could be true?

5.

6.

7.

(A)

(B)

(C)

(D)

x = c is a vertical asymptote of the graph off

lim [Cx) =t-= fCc) x-->c

The graph of [has a cusp at x = c.

f(c) is undefined.

1:12 cos t dt =

(A) -srnx (B) -sin x - 1

(C) sin x - 1 (D) 1 - sin x

If x 3 + 2x 2y - 4y = 7 then when x = 1 dy = . , '. > dx

(A) -8 (B) 9 2

(C) -~ (D) 7 2

y

~"~~"~ __ __ """J/ ""'~~~~- -~-~;",,~

~~"'~"'~ -~'~~"'" \~,\"",~ ~,-~~,I'"

\"","~, ,""'~/JJ ~ \\", .... ," """,,11 \ \, '\ ' ''', .... '-".I" / IJIII 1' \ \''''"', .... ,~/'/')JJI 11 ~\,\ , " ..... ,.,/./",/1111 I l j ,""J~ .... ,"\,\\\ I.X

; ~ ~ ~ ~; ~ ~;; -::~~ ~ ~ ~ ~ ~ ~ : 1 11 /1"//-' - .... ''''~\'l' 11 1 '/'/'''' .... ,", .... ,'" II"~""'~~ ,~"""'" """,,~- .... "~" "\ ,

"""'~-- --,-""" """_,~- -~ ........ "" ' ~

8. Shown above is a slope field for which of the following differential equations?

(A) dy

(B) dy x - =x- y -=-

dx dx Y

(C) dy

(D) dy

- = x+ 1 -=xy dx dx

222 PART III: PRACTICE EXAMINATIONS ¥ '.

Copyright © 2017 Pearson Educatiq}l, Inc.

9.

10.

11.

12.

I, 1 + sin 3x 1m - =

X~'TT/2 1 - cos 4x

9 3 (A) -- (B)

J6 4 (C) 0 9

(D) 16

What is the instantaneous rate of change at x = 3 of the function f x 2 - 2

given by f(x) = x + I ' ?

(A) _12 (B) _1 (C) 13 (D) 17 16 8 16 16

If [is a linear function where 0 < a < band -m is a nonzero constant,

then f: rex) dx =

(A) 0

(A.) In 9

(C) 3ln 3

(B) ab 2

(C) ( b) (D) a2 - b2

m a - 2

forO < x S 3 c . ) then lim f(x) is lor 3 < x ~ 4 ~3

(B) 1n 27

(D) nonexistent

y

3.

-4

13. The graph of the function f shown in the figure has a horizontal tangent at the point (-1, - 3) aDd a cusp at (2, 0). For what values of x, -5 < x < 5) is [not differentiable?

PART III: 'PRACTICE EXAMINATIONS

(A) a only

(C) -1 and 0 only

(B) 0 and 2 only

(D) -1,0, and 2

Copyright © 2017 Pearson Education, Inc.

223

224

14. A particle moves along the x-axis with velocity given by vet) ,-- 3t 2

1- st - 2 for t .2 O. If the particle is at position x ::- 3 at time t == 0, wl1at is the position of the particle at time t .....; l?

(A) 1.5 (B) 4.5 (C) 6

15. If F(x) = 1;\/t2 + 3 dt, then P'(2) =

(A) V7 (D) 4v7 (C) 2V19"

.16. If/(x) = cos (e 2x), thenJ'(x) =

(A) - ·2 sin (e 2x)

(C) 2 sin (e 2x)

y

(B) -2e 2x sin (e~

(D) sin (e4

2 ..

(D) 11

(D) 4V19

17. The graph of the function f for 0 < x $ 4 is shown above. Of the following, which has the greatest value?

(A) Trapezoidal sum approximation of 104 f(x)dx with 4 subintervals

of equal length

(B) Right Riemann sum approximation of J: f(x)dx with 4 subintervals of equal length

(C) Left Riemann sum approximation of J04 f(x)dx with 4 subinter

vals of equallcngth

(D) J; f(x)dx

18. An equation of the line tangent to the graph of y = 3x - cos x at x = 0 is

(A) y = 2x .

(C) Y = 3x + 1

(B) y = 2x' - 1

(D) y = 3x - 1

P/\RT III: PRACTICE EXAMINATIONS ------------------------------------------

Copyright © 2017· Pearson Education" Inc:

· . ,

19~ Iff"(x) =-= (x - l)(x .[ . 2)3(X - 4)2, then the graph offhasinflec:tion points when x =

(A) 1 only

(C) -2 and 1 only

(B) I and 4 only

(D) -2,l,and4

20. If 12-3 f(x)dx = -13 and 15-3

f(x)dx = -10, what is the value of

1~f(X)dx?

21.

(A) -23 (B) -3

(C) 3 (D) 23

tiy . ' If dt = my and m is a nonzero constant, then y could be

(A) 4e mty

(C) emt + 4

(B) 4e mt

(D) ; y2 + 4

y

22. The graph of the function f shown above has horizontal tangents at , 5 • x

X = -1 andx = 3' Letg be the functJOn defined by g(x) = 10 f(t)dt. For what vCl,lues of x does the graph of g have a point of inflection?

,(A) 0 only (B) ! only

(C) 3 only (D) -1 and l .3

23. The minimum acceleration attained on the interval 0 < t ~ 4 by the particle whose velocity is given by v(t) = t 3 - 4t2 - 3t + 2 i,s

(A) -16 (B) -10 (C) 25 (0)-:3 3

I\RT III: PRACTICE EXAMINA nONS 22~


226

1

.\ I

----------------------------~

y

24. The graph Off is shown in the figure above. Which of the foUowing could be the graph of the derivative off?

(A) y (B) y

(C) y (D) y

25. What is the area of the region between the graphs of y == x:' and y = --:.X. - .1 from x = ·Oto -x. == 2?

(A) 0 (B) 4 (C) 5 CD) 8

I ;'Cx) I = ~ I ~ I ~ I ~ I ~ I 26. The polynomial function f has selected values of its first derivative l'

given in the table above. Which, of the following statements must be true?

(A) f changes concavity at least twice on the interval (-1,3).

(B) f has a local minimum at x = L

(C) f is increasing on the interval (l) 3).

(D) f has a local maximum at x =- 2.

PART III: PRACTICE EXAMINATIONS

Copyright © 2017 Pearson Educatfo'n, Inc.

I

27. What is the average value of y = x 3 Vx4 -t~ on the interval (0, 2]?

(A) ~ (B) . 49 .(C) .125 (D) 49 3 3 12 6

28. If I(x) = tan (3x), then f' CJ) =

(A) .1-3

(B) 4 (C) 12 (D) 6\13 .

' --

29. The side of a cube is increasing at a constant rate of 0.2 centimeter per second. In terms of the surlace area 5, what is the rate of change of the volume of the cube, in square centimeters per second?

(A) 0.15 (B) 0.25 (C) 0.65 (D) 0.045

X(I)

30. A particle moves along a straight line. The graph of the particle's position xU) at time t for 0 < t < 8 is shown above. The graph has horizontal tangents at t = 1 and t = 1; and a point of inflection at t = 13°,

For what values of t is the velocity of the particle decreasing?

PART III: PRACTICE EXAM/NA j IONS

(A) o. < t < 13°

(C) 4 < t < 7

(B) 1 < t < lZ 3

(D) 1.Q < t < 8 3

~ End of Part A of Section I <}


227

Calculus AB-Exam 1 Section I, Part B

Time: 45 minutes Number of questions: 15

228

A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONSIN THIS PART or THE EXAMINATION.

Directions: Solve each of the following problems. After examining the-form of the choices, decide which is the best of the choices given.

In this test:

1. The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value.

2. Unless otherwise specified, the domain of a function fis assumed to be the set of aU real numbers x for which I(x) is a real number.

y

31. The graph of a function f is shown above. Which of the following statements about I is false?

(A) lim f(x) exists. x--a

(B) fhas a relative minimum at x = a.

(C) f(a) exists.

(D) f is can tinuous at x = a.

32. Let I(x) = 2e 3x arid g(x) = 5x 3. At what value of x do the graphs off

and g have parallel tangents?

(A) -0.366 (B) -0.344 (C) -0.251 (D) -0.165

"' . Copyright © 2017 Pearson Education( hie.

i !

Y Y

-J.--+---I-------.....l~ X

a b

Y'" g'(x) y=h'(x)

y = J'(x)

33. The graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative minimum on the open interval a < x < b?

34.

(A) g only

(C) f and h only

(B~ h only

(D) f, g, and h

. 2 2 The first derivative of the function f is given by I' (x) = Sl~ X - g' How many critical values does fhave on the open interval (0, IO)?

(A~ Two

(C) Four

(B) Three

(D) Six

35. Letfbe the function given by f(x) = x2l3. Which of the following statements about fare true?

I. f is continuous at x ;;...:: O.

II. fis differentiable at x = o. III. fhas an absolute minimum at x = o.

(A) I only (B) II only

(C) I and II only (D) I and III only

36. If f is a cqntinuous function and if F' (x) = f(x) for all real numbers x, then I-I f(3x)dx ~

(A) 3F(2) - 3F( -1) 1 1 (B) -P(2) - - F(-I) 3 3

(C) 3F(6) - 3F(- 3) 1 1 (D) - F(6) - - F( -3) 3 3

PART II~: PRACTICE EXAMINATIONS 229

Copyright © 2017 Pearson Education. Inc.

230

3 J

37. If a * 0, then lim X6 - 4(, is x->a a - x'

(A) nonexistent

(C) 0

J (ll) --

2(.]3

(D) _1 2a3

-'---

38. Popl,1lation P grows according to the equation ~ - kP, where k is a

constant and t is measured in years. If the population doubles every 12 years) then the value of k is

(A) 0.058 (B) 0.279 (D) 1.792

x 1 3 6 9

f(x) 15 25 40 30

3.9. The function f is continuous qn the closed interval [1,9) and has values that are given in the table above. Using the subintervals [1,3],

[3, 6] i and [6, 9 J, what is,the trapezoidal approximation of J: f(x)dx?

(A) 110·

(C) 242.5

3

y

(B) 175

(D) 262.5

40. The base of a solid is a region in the first quadrant bounded by the x-axis, the y~axis, and the line x + 3y =- 9, as shown in the figure above. If cross sections of the solid perpendicular to the y-axis are isosceles right triangles with the hypotenuses in the xy-plane, what is the volume of the solid?

(A) 6.75 (B) 13,5 (C) 20.25 (D) 40.5

PART III : PRACTICE ;EXAMINATIONS ----------------------------------------------

Copyright © 20] 7 Pearson Edud'ri9n, Inc.

-'

, .i

41.

----------

Which of the following is an equation of the line tangent to the graph of f(x) = x 6

- X4 at the point where f' (x ) ---: -1?

(A) y - -x - 1.031

(C) Y = -x + 0.934

(B) Y = . -x - 0.836

(D) y = -x + 1.031

42. Let G(x) be an antiderivative of f(x).lf G(2) = 3, then G(8) =

(A) 1'(8) (B) 3 + 1'(B)

(C) 1:0 + f(t))dt CD) 3 + f: f(t)d~ 43. A particle moves along a straight line with velocity given by

vet) = 5 + l.492- cl at time t > O. What is the acceleration of the particle at. time t = 2?

(A) -0.196 (B) 0.7433 (C) 5.041 (D) 11 .205

44. Let f be a function that is differentiable on the open interval (-3, 7). If f(-I) = 4, f(2) = -5, and f(6) = B, which of the following must be true?

L For some c, -1 < C < 2, p(e) = -3.

II. f has a relative minimum at x = 2.

III. For some e, 2 < c < 6, fCc) == ,4.

(A) I only

(C) I and III only

(B) I and II only

(D) 1,'II, and III

45. If 0 <; k < ; and the area under the curve y =- sin x from x = k to

x .; ; is 0.75, then k =

(A) 0.253 (B) 0.723 (C) 0.848 (D) 1.318

PART III : PRACTICE EXAMINATIONS 231 ----------------------------------------------- -----Copyright © 2017 Pearson Education, Inc.

Time: 30 minutes Number of problems: 2

232

Calculus AB-Exam 1 Section II) Part A

------------------------------

A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS IN THIS PART OF THE EXAMINATION.

y

1. Let f and g be the functions defined by

f(x) = O.5i/ - 2.2x3 + 4x + 2.75 and g(x) = 2.75 C05(;). Let R

andS be the two regions enclosed by the graphs of f and g shown in

the figure above.

(a) Find the sum of the areas of the regions Rand S.

(b) Find the volume of the solid generated when R is rotated about the x-axis.

(c) The region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid.

Copyright © 2017 Pearson EducatloD, Inc.

f: 1--

~.

2. For 0 S t < 4, a particle is moving along the x-axis. The velocity of tue particle is given by y(t) :....: 3 cos(erl2 ) +, 1.5. The acceleration of the particle is given by a(t) := - telJ2 sinCel12) and x(O) = 2.7.

, ~RT III; PRACTICE EXAMINATIONS

(a) Is the speed of the pa-rticle increasing or decreasing at time t = 2.4? Give. a reason for your answer.

(b) Find the average velocity of the particle for the time period o < t < 4.

(c) End the total distance traveled by the particle from time t = 0 to t = 4.

(d) For 0 < t S; 4, the particle changes _direction twice. Find the position of the particle at the time when it changes from moving left to moving right.

~ End of Part A of Section II ~

233


Time: 60 minutes Number of problems: 4

234

Calculus AB-Exam 1 Section II) Part B

NO C;:ALCULATOR MAY BE USED IN TIllS PART OF THE EXAMINATION.

T (days) 0 1 3 4 6

h(t) (em) 5.1 6.8 11 13.5 20.1

3. Jill bought an amaryllis plant hoping it would soon bloom. She decided to' chart its 'growth before it flowered. The length of a leaf is modeled by a differentiable'function h that is increasing and concave up for o < t < 6. The table above gives selected values of h(t), where t is measured in days and h(t) is measured in centimeters.

(a) Use the data in the table to estimate hi (3.5). Show the computations that lead to your answer.

(b) Using correct units, explain the meaning of ~f: h(t)dt in the con

text of this problem. Use a right Riemann sum with the four

subintervals indicated by the table to estimate if: h(t)dt.

(c) Is your approximation in part (b) grea ter than or less than

~J: h(t)dt? Give a reason for your answer.

(d) Evaluate J: h'(t)dt. Explain the meaning of this expression.

Copyright © 2017 Pearson Edl)c1itl9n. Inc . •

I i

I

t •

·~

y

4. The figure above shows the graph of f', tl1.e derivative of a twicedifferentiable function f, on the interval [-.2,3]. The graph of f' has horizontal tangents at x ~ -2, x = -0.8, x = lA, and x = 3. The areas of the regions bounded by the x-axis and the graph of J' on the intervals [-2,0] and [0,3) are 2 and 7 respectively .

PART III: PRACTICE lXAMINATIONS

(a) Find all x-coordinates at which f has a relative minimum. Give a reason for your answer.

(b) On what open intervals contained in -2 < x < 3 is the graph of

f both concave up and increasing? Give a reason for your answer.

(c) Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer.

(d) Given that f(O) = 9, write an expression for f(x) that involves an integral. Find f(-2) and f(3) .

235


236

S. Consider the curve given by x 2 + 3l = 1 + 3xy. It can be shown dy 3y - 2x

that ~A~ = 6 3' (./.A Y . . x

(a) Write an equation for the l..ine tangent to the curve at the point (1,1).

(b) Fi.nd the coordinates of all points on the curve at which the line

t~gent to. the curve at that point is vertical.

d~y' (c) Evaluate cb:2 at the point on the curve where x = 1 and y = 1.

PART III: PRACTICE EXAMINATIONS ... Copyright © 2017 Pearson Educatiqn, Inc.

6. Consider the differential equation ix ::: x 2(2y + 1).

PART III : PRACTICE EXAMINATIONS

(a) O~] the axes provided, sketch C1 slope field for the given differential -equation at the nine points inclicated.

y

• •

--~--~----~x -1 I

• •

(b) While the slope field in part (a) is drawn at only nine points, it is defined at every point iIi the X)'-plane. Describe all points in the xy-plane for which the slopes are positive.

(c) Find theparticular solution y = f(x) to the given differential equation with the initial condition f(O) = 5.

237 -------------------------------Copyright © 2017 Pearson Education. rnr

•

Problem

1. 2.

3. ,4.

5.

6. 7.

8.

9. 10.

II. 12.

13. 14.

15.

16. 17. 18.

19. , 20.

2l.

22.

23. 24.

25.

26. 27. 28.

29.

30.

Practice Examln,atiollS Calcullus AB~Exam 1 Section I

Part A-No Calculator

Answer Key Concept (A) Differentiability' (B) Graph analysis for extrema (D) Defini te, in tegral (C) Continuity/Differentiability (C) Definite· integr:at ' (B) I.r:nplicit differentiation (A) Definite integral (B) Slope field (D) L'Hospitars Rule (D) Instantaneous rate of change (A) Relationships of derivatives (D) Limit (B) Graph analysis for differentiabiJjty (B) Velocity/Position (D) Fundamental Theorem of Calculus (B) Derivative with Chain Rul~ , (C) Area approximations (D) Tangent line (C) Points of inflection (B) Integration rules (B) Exponential growth derivati~e CD) , Graph analysis for points of inflection (C) Velocity/ Acceleration optimiza tion (A) Derivative graph (D) Area between curves (A) Analysis of derivative data (D) Average value (C) Numerical derivative (A) Rate of change' (A) Graph analysis for PositonJVelocity

Copyright © 2017 Pearson Education, Inc .

401

Part B-CalcuIator Allowed

Problem Answer

31. CD) 32. (A) 33. (C)

34. (A) 35. (D) 36. (D) 37. (B) 38. (A) 39. (C) 40. (C) 41.- (A) 42. (D) 43. (A) 44. (C) 45. (B)

402

Key Concept Graph analysis Nwneri-cal derivative with calculator Derivative graph analysis for extrema Critical values Continuity/Differen tiability Fundainental Theorem of Calculus Limit Exponential growth derivative Trapezoidal Rule Volume by cross section Tangent Une-Fundamental Theorem- of Calculus Velocity/Acceleration Mean Value-Theorem/Intermediate Value Theorem Area under curve

Copyright © 2017 Pearson EdU~;A(on, Inc . . J

ANSWERS AND SOLUTIONS

.. _____ ... ... r;.,,"

Calculus AB~Exam 1: Section It, Part A

1. f(x) = g(x) at x = 0, x = U\ 1 09294, and x = 3.7724666. Let A = 1.8109294 a'nd B = 3.7724666

A 'B ' (a) Area·= 10 U(.x) -:- g(x)]dx + fA [g(x) - f(x)]dx = 9.931

(b) Volume = 1T f:((f(x»2 . - (g(x)2Jdx = 64.325 , '~ " .

(c) Volume = fA (g(x) - f(x»)2dx = 31.790

2. (a) v(2A) ~ -1.452 < 0 and a(2.4) ~ 0.884 > O. Since the velocity and acceleration have , opposite signs, the speed is decreasing at t = 2A.

(b) Averagevelocity = i lo\(t)dt = 1.158

(c) Total Distance = f041 vet) j dt ~ 7.309

Cd) The particle changes from moving Jeft to moving right when the veloCity changes from negative to positive. .

(2.865 x(2.865) = 2.7 + J 0 v(t)dt = 3.967

3. (a) h' (3.5) ~ h( 4l = ~(3) c-= 13.5 ; 11 = 2.5 em/day

(b) i f~ h(.t)dt is the average length of a leaf in centimeters during the first six days of the

data collection.'

i J: h(t)dt ~ i [1(6.8) + 2(11) + 1(13.5) -I- 2(20.1») '-' 8~.5 = 13.75 em

(c) Since the graph of h( l) is strictly increasing on (0, 6], the right Riemann sum is an overestimate, and therefore greater than the true area.

Cd) I: h'(t)dt :-:= h(6) - h(O) = 20.1 - 5.1 ::: 15. The leaf grew a total of 15 em in the first

six days.

ANSWERS AND SOUTIONS 403

Copyright © 2017 Pearson Education. Inc.

404

CalCulus AB-, Exam 1: Section II, Part B

4. (a) f' (x) :-: 0 at x =-: -2, x = 0, and x = 3. x == 0 is tl1e only critical point where, f' chan

5.

from negative to positive. Therefore, f has a relative ~inimUm at x = 0:

(b) f is concave up when f" is positive, which is where f' is increasing. f is increasing where; f' is positive. Therefore, f is both concave up and increasing on 0 < x < 1.4 because l' is both increasing and positive on this interval.

ec) The graph of f has inflection points at x "- -0.8, where f' changes from decreasing to increasing and at x = lA, where f' changes from increasing to decreasmg.

Cd) f(x) = 9 + f;f'(t)dt '

r-2 f ( - 2) == 9 + J 0 f' (t) dt = 9 + 2 = 11

1(3) == 9 + f~ f'(t)dt == 9 -I- 7 = 16

dy I 3 - 2 l' 1 (a) dx (1,1) = 6 - 3" = '3 => y - 1 = "3 (x - 1)

(b) 6y - 3x = 0 ~ 2y = X

(2y)2 + 3~ = 1 + 3(2y)y

4y + 31 == I + 61 y2 '= 1

y=±1=>x==:+:2

(-2, -1) and (2,1)

d2y dY

(3Y

_ 2X) (6y - 3X)(3 ~~ - 2) - (3y - 2x)(6~, ~ 3) (c) dx-i = dx 61 - 3x :-:, (6y - 3x)2

(6 - 3{ 3· t - 2) - (3 - 2)(6. t - 3) (6 - 3i

" j . ,

Copyright © 2017 Pearson EducaL\~\l , Inc.

3(-1) - )(-1) 2 = 9 =-9


- It

6. (a)

(b) Slopes are positive for pojnts where x ~ 0 and y > -t. j' dy _/ 2

(c) 2y + 1 - x dx

If 2dy / 2 "2 2y + 1 = x dx

± In 12y + 11 = t ~ + C

In 12y + 11 = t ~ + c 12y + 11 = e ~ ; + C

12y + li.= : Ce~X> 12(5) '+ 1 j = CeQ

ll=C

2y + I = lle~X>

2y = lle~~ -: 1

y == ±( lle~X> - 1)


Copyrigbt © 2017 Pearson Education, Inc.

405

Calculus AB-Exam 1 - Ms. Orloff

Documents

Transcript of Calculus AB-Exam 1 - Ms. Orloff