Page 14 - 2024-calc4e-SE proofs-4e.indd

Prepare and practice for the AP ® Calculus exams.

Sullivan 10 apcalc4e 45342 ch08 620 705 2pp August 28, 2023 15:50
© 2024 BFW Publishers PAGES NOT FINAL - For Review Purposes Only - Do Not Copy.
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AP Practice Exam: Calculus AB
AP Practice Exam: Calculus AB Preparing for the AP Exam
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Make sure you’re prepared.
Section 1: Multiple Choice, Part A
A calculator may not be used for Part A. A full-length AP ® Practice Exam
1. If f (x) = e 4x + sin(2x), then f ′ (0) = 8. The function f is differentiable and its derivative is continuous for Calculus AB after Chapter 8
on the interval (−4, 5). The table below lists several values of
(A) 1 (B) 2 (C) 4 (D) 6
f and f ′ in the interval.
(x 2 − 1) f (x) and for Calculus BC after Chapter
2. If lim f (x) = 3, then lim =
x→1 x→1 x − 1 x −3 −1 0 3
(A) 0 (B) 4 (C) 6 (D) Does not exist. 2 10 each contain 45 multiple-
f (x) −16 2 2
3. The graph of the function f is shown below. Which of the 3 choice questions and 6 free-
following statements is false? f ′ (x) 3 −1 0 15
y � 3 response questions. These exams
Then −1 f ′ (x) dx =
2
4
(A) −16 (B) (C) 12 (D) 16 match the AP ® Calculus exams in
3
1
√
9. If y = 4x + 6e tan x , then y ′ equals length and scope.
� 1
(A) 4 + 6e tan x sec 2 x (B) √
1 2 3 4 5 6 x 2 4x + 6e tan x
2 + 3e tan x 2 + 3e tan x sec 2 x
(A) lim f (x) = f (2) (C) √ (D) √
x→2 − 4x + 6e tan x 4x + 6e tan x
(B) The function f is discontinuous at x = 3. Sullivan 12 apcalc4e 45342 ch10 778 906 2pp September 5, 2023 16:29
10. The function f is continuous on 1 ≤ x ≤ 5 and
(C) lim f (x) exists. differentiable on 1 < x < 5. If f (1) = 10 and f (5) = 50,
= 50
50
x→4
=
(D) The function f is continuous at x = 5. then the Mean Value Theorem guarantees that 902 Chapter 10 • Infinite Series
(A) f is linear on the interval 1 ≤ x ≤ 5.
4. Let f be the function f (x) = x 3 − 15x 2 − 1800x + 2000.
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On which interval is the function f both decreasing and (B) f ′ (c) = 10 for at least one c between 1 and 5. AP Practice Exam: Calculus BC Preparing for the AP Exam
concave down? (C) f ′ (c) = 0 for at least one c between 1 and 5.
(A) x < −20 (B) −20 < x < 5 (D) f (c) = 30 for at least one c between 1 and 5.
Section 1: Multiple Choice, Part A
(C) 5 < x < 30 (D) x > 30 1 � 1 2 3 � A calculator may not be used for Part A.
11. The Riemann sum e / 20 + e / 20 + e / 20 + ··· + e 2 e e is an
dy 20
5. If y = e 2x cos(3x), then = 3x 2 − 7x − 4
dx approximation for which integral? 1. dx = 7. Find y ′ at the point (3, 6) on the graph of y 2 = 2x 2 + xy.
� 2 � 2 x − 3
(A) −6e 2x sin(3x) x 1 6 7
(A) e / 20 dx (B) e x dx 3 (A) (B) (C) (D) 2
(B) e 2x (2 cos(3x) − 3 sin(3x)) 0 0 (A) x 2 + 2x + 2 ln |x − 3| + C 6 5 4
2
� 2 � 2
(C) e 2x (2 cos(3x) + 3 sin(3x)) 1 1 x 8. The function f has a second derivative given by
(C) e x dx (D) e / 20 dx 3 √
(D) e 2x (cos(3x) − sin(3x)) 20 0 20 0 (B) 2 x 2 + 2x − 10 ln |x − 3| + C f ′′ (x) = x 2 (x − 1) x + 1. At what values of x does f
have a point of inflection?
area
the
the
of
W
quadrant
hat
What is the area of the region in the first quadrant bounded by
is
by
first
in
bounded
the
region
6. Let f be the function defined below. For what value of k is f 12. What is the area of the region in the first quadrant bounded by (C) 1 (3x + 2) 2 + 2 ln |x − 3| + C (A) 0 only (B) 1 only
continuous at x = 0? the graph of y = 1 + e −2x and the line x = 3? 2
 (C) 0 and 1 only (D) −1, 0, and 1
sin(7x) 1
 for x < 0 (A) 2 − 2e −6 (B) 3 − e −6 (D) 3x + 2 + 2 ln |x − 3| + C
2x 9. If the function f is continuous and if F ′ (x) = f (x) for all
f (x) = 2
x 2 5
1
 real numbers x, then 2 f (3x + 2) dx equals
k + 2 ln(x + e x+1 ) for x ≥ 0 7 − e −6 (D) 5 − 2e −6
(C) 2. e 3t dt =
3 3 7 2 2 0 (A) 3F(5) − 3F(2) (B) 1 F(5) − 1 F(2)
(A) − (B) −1 (C) (D) � 4 1 3 3
2 2 2 √ 1 + x (A) e 3x 2 − 1 (B) 3 e 3x 2 − 1
equal
13. Using the substitution u = x, √ dx is equal to tois is equal to 1 1
� 1 1 + x 3 (C) F(17) − F(8) (D) 3F(17) − 3F(8)
5
7. Let f be the function defined by f (x) = (x − 3) 4 for all x. which of the following? 3 3
� 4 � 2 (C) 2xe 3x 2 1 (2x) e 3x 2 ∞ 3 k+2
Which of the following statements is true? 1 + u 2 1 + u 2 (D) − 1
(A) du (B) du 3 10. What is the sum of the series k=1 5 k+1 ?
Sullivan 12 apcalc4e 45342 ch10 778 906 2pp September 5, 2023 16:29 (A) f is continuous and differentiable at x = 3. 1 1 + u 1 1 + u 3. What is the slope of the tangent line to the graph of
(B) f is continuous but not differentiable at x = 3. � 4 u + u 3 � 2 u + u 3 3 5 27
(C) f is differentiable but not continuous at x = 3. (C) 2 du (D) 2 du y = e 6x − sin(4x) at x = 0? (A) 2 (B) 2 (C) 2 (D) 10
(D) f is not continuous and not differentiable at x = 3. 1 1 + u 1 1 + u
(A) 0 (B) 1 (C) 2 (D) 10 11. The base of a solid S is the region enclosed by the graph
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AP Practice Exam: Calculus BC 905 2e 5x − 5e 2x + 3 of y = √ x, the line x = 3, the line x = 1, and the x-axis. If
4. lim = the cross sections of S perpendicular to the x-axis are
x→0 x 2 − 2 cos(2x) + 2
squares, then the volume of S is
38. Which of the following differential equations for a population P 45. The function f is continuous on the closed interval [1, 3]. (A) 0 (B) 3 (C) 3 (D) 3 √ 4 √
could model the logistic growth function shown in the figure If f (1) =−2 and f (3) = 4, then the Extreme Value 4 2 (A) 2 (B) 2 3 (C) 4 (D) 3 3 3 − 1
below? Theorem guarantees that 5. Which of the following series converges? 12. Let f be the function given by f (x) = x x . If three
P(t) ∞ 1 subintervals of equal length are used, what is the value of the
(A) f (c) = 0 for at least one c in the interval [1, 3]. Sullivan 12 apcalc4e 45342 ch10 778 906 2pp September 5, 2023 I. 16:29 Left Riemann sum approximation for 1.2 x x dx?
1.8
k=1 k 2/3
500 (B) f ′ (c) = 0 for at least one c in the interval [1, 3]. (A) 5.0(1.2 1.2 + 1.4 1.4 + 1.6 1.6 )
k
∞
(C) f ′ (c) = 3 for at least one c in the interval [1, 3]. II.
k=1 k 3 + 5k + 1 (B) 0.2(1.2 1.2 + 1.5 1.5 + 1.8 1.8 )
250 (D) f attains an absolute maximum value f (c) ≥ f (x) 906 Chapter 10 • Infinite Series (C) 0.2(1.2 1.2 + 1.4 1.4 + 1.6 1.6 )

for all x in the interval [1, 3]. III. ∞ (−1) k ) 1
(
k k=1 = 1 k k (D) 0.2(1.4 1.4 + 1.6 1.6 + 1.8 1.8 )
x
f (t) dt. Find
(B) I and III (a) Let g be a function given by g(x) =
t Section 2: Free Response, Part A Section 2: Free Response Part B (A) I and II (B) I and III 13. The volume of a box with a square top and bottom is to
(A) I and II
0
be k cubic inches. If a minimum amount of cardboard is
A graphing calculator is required for Part A. No calculator is allowed for Part B.
to be used to construct the box, what must be the area, in
(C) II and
(D) III only
d P d P (C) II and III III (D) III only sin(2x) − g(x)
square inches, of the top of the box?
(A) = 2P − 0.004P 2 (B) = P − 0.004P 2 3. Water is dripping from a faucet into a 2-gallon bucket that has lim e 3x − cos(2x)

x→0
x
4
−
dt dt 1. The figure below shows the graphs of height h 24 in. The height of the water in the bucket 2 2 at 4x − 1 1 dx dx = = (A) 3 √ − 3x (B) 2 k 3 √ (C) 3 k 2 (D) 4 3 k 2
6. 6.
k
1
x
2x 2 − x − 1
x
−
−
d P d P r 1 (θ) = 2 + 3 sin θ + 4 cos θ and r 2 (θ) = 2 + 6 cos θ time t, 0 ≤ t ≤ 60, can be modeled by a differentiable function h, Show the work that leads to your answer.
(C) = 2P 2 − 0.004P (D) = P 2 − 0.004P π where h is measured in inches and t in minutes. The table below 14. Let P(x) = a + b(x − 1) + c(x − 1) 2 + d(x − 1) 3 be the
(A) ln |(x − 1)(2x + 1)| + C C
(A) ln |(x − 1)(2x + 1)| +
third-degree Taylor polynomial for the function f
dt dt on the interval 0 ≤ θ ≤ . Let S be the shaded region (b) The function y = f (x) is the particular solution to the
2 shows the height of the water at select times t. about x = 1. A table of values for f (k) (1) is given below.
y 2
(B) ln (x − 1)(2x + 1) 2 + Cln (x − 1)(2x + 1) 2 +
39. Find the volume of the solid obtained by revolving the curve bounded by the two graphs, the x-axis, and the y-axis. (B) differential equation dy = Find c + d. initial condition
with
√ 3 cos x π The two curves intersect at point P. t (minutes) 0 10 20 30 40 50 60 dx x + 1 k 0 1 2 3
(C) 2 ln |(x − 1)(2x + 1)| +
y = x + from x = to x = 3π about the x-axis. (C) 2 ln |(x − 1)(2x + 1)| + C f (0) = 2. Use Euler’s method, starting at x = 0 with two
x 2 y h (inches) 0 9.4 15.2 18.6 20.8 22.0 22.8 f (k) (1) 4 4 −2 3
1
(A) 41.120 (B) 52.127 (C) 129.183 (D) 142.093 8 (D) 2 ln x − 1 steps of equal size, to approximate 3 f . 1 Show the
2x + 1 + C+ C

(D) 2 ln
x − 1
2x + 1
(a) Using the data above, approximate h ′ (25). Show the (A) − 2 2 (B) − 2 (C) 0 (D) 1
40. A particle moves in the xy-plane so that its position at any time t r 1 (θ) 5 2 1 3sin θ 1 4cos θ computations that lead to your answer.
6 computations and explain the result, including the units.
is given by x(t) = t 2 + 3t and y(t) = 2t 3 − 21t 2 + 60t. What is
60
the speed of the particle when t = 3? 4 P (b) Approximate 0 h(t) dt with a trapezoidal sum with (c) Use separation of variables to find y = f (x), the
r 2 (θ) 5 2 1 6cos θ three intervals of equal length. Use the result to particular solution to the differential equation
(A) 9 (B) 12 (C) 15 (D) 21
2 S 1 60 y 2
x calculate h(t) dt, and interpret the meaning of dy
41. Let g be a differentiable function and G(x) = g(t) dt. The 60 0 = with initial condition f (0) = 2.
0 dx x + 1
table below gives a value of G and values of the first two 2 4 6 8 x the result. Be sure to use correct units. Reviewed by the experts.
1
derivatives of G. Use a second-degree Taylor polynomial for the (c) If the function that models h is h(t) = 24 − 24e −0.05t , 6. The function f is defined by f (x) = (1 + x 2 ) 2 . The Maclaurin
0.8 Every AP ® Practice Exam, in the section and
function G about x = 1 to approximate g(t) dt. (a) Write, but do not evaluate, an expression involving one or determine the concavity of h. Does the trapezoidal sum series for f is given by
0 found in (b) overestimate or underestimate the height of
6
4
2
more integrals that gives the area of S. chapter reviews and practice exams, has
2n
n
1 − 2x + 3x − 4x + ··· + (−1) (n + 1)x
+ ···
x 1 the water in the bucket?
π
been conceived of and carefully checked by
G(x) 3 (b) Find the angle θ in the interval 0 ≤ θ ≤ that (d) Is there a time t between t = 40 and t = 60 minutes at (a) Use the series to find each of the following:
G ′ (x) 6 corresponds to the point on the curve 2 which h ′ (t) = 0.10? Justify your answer.
i. the value of f (4) (0)
G ′′ (x) 5 experts in AP ® Calculus to ensure that they
r 1 (θ) = 2 + 3 sin θ + 4 cos θ 4. Let f be a function defined on the closed interval 0 ≤ x ≤ 16
ii. the coefficient of the x 22 term
(A) 1.9 (B) 2.0 (C) 4.3 (D) 9.4 with f (0) =−3. The graph of f ′ , the derivative of f , consists offer comprehensive support to your
that is furthest from the origin. Justify your answer. of 4 line segments as shown below. Also shown is the graph
42. A particle travels along a straight line with a velocity (b) Find the radius of convergence for the Maclaurin series
(c) The radial distance between the two curves changes for 4 exam prep.
for f . Show the work that leads to your answer.
which intersects the graph of f ′ at points P and Q.
5 √ π of y =
of v(t) = sin(2π t) feet per second. What is the 0 ≤ θ ≤ . Suppose θ changes at a constant rate of x (c) Let P(x) be the fourth-degree Maclaurin polynomial
t − 10 2
total distance, in feet, traveled by the particle during the dθ = 1 for all times t ≥ 0. Find the rate at which the y for f (x). Use the graph of f (5) shown below together
xxviii
with the Lagrange error bound to show that
time interval 0 ≤ t ≤ 9 seconds? x x v iii dt 3 © 2024 BFW Publishers PAGES NOT FINAL 1
P
2
(A) 2.945 (B) 5.497 (C) 7.006 (D) 10.994 distance between the two curves changes for θ = π . | f (0.5) − P(0.5)| < 10
6 For Review Purposes Only, all other uses prohibited
Show the work that leads to your answer.
1
Q
43. A particle travels along a straight line with velocity v(t). The 2. At a large distribution center, workers process orders during the Do Not Copy or Post in Any Form.
table below gives the values for v(t) and v ′ (t) for selected times. five-hour shift from noon to 5:00 p.m. Orders are received at a y
√
At which of the given values of t is the speed of the particle rate of r(t) = 400 9 + 4t − t 2 orders per hour for 0 ≤ t ≤ 5 4 8 12 16 x 100 (0.684, 90.243)
increasing? where t is the number of hours since noon. During the same time
t 2 4 6 8 interval, orders are processed and packaged by the workers at a Graph of f 9 0 0.2 0.4 0.6 0.8 1.0 x
constant rate of 1200 orders per hour. At the beginning of the
v(t) −5 6 −9 18 01_apcalc4e_45342_fm_i_xxix_3pp.indd 28 (a) Determine whether f has a relative minimum, relative 2100 10/11/23 2:39 PM
v ′ (t) −3 −8 6 10 shift, there are 2000 orders that had not been processed and maximum, or neither at x = 12. Justify your answer.
packaged.
(A) 8 only (B) 4 and 6 (C) 4 and 8 (D) 2 and 8 (b) Let g be a function given by g(x) = ( f (x)) 2 . Write an 2200
(a) Using correct units, find and explain the meaning of equation of the line tangent to the graph of g at x = 8.
44. Sand is poured through a chute onto a conical pile at the rate of 4 r(t) dt. Express your answer to the nearest integer. (c) Let h be a function given by h(x) = xe −0.25 f (x) . Find all 2300
25 cubic feet per minute. The bottom radius of the conical pile is 3 (0.202, 2357.685)
the critical numbers of h. 2400
always half the height. How fast does the radius of the base (b) Write an equation involving an integral expression of r(t) 8
change when the pile is 10 feet high? The volume of a cone is that gives N(t), the number of unprocessed orders at (d) Find 4 xf ′′ (x) dx. Graph of f (5)
time t. 5. Consider the graphs of the function y = f (x) and the line
1
2
V = πr h tangent to the graph of f at x = 0 pictured below. (d) Find the first four nonzero terms and the general term for
3 (c) At what time t, for 0 ≤ t ≤ 5, is the number of x
(A) 0.010 (B) 0.040 (C) 0.159 (D) 0.318 unprocessed orders a maximum? y y 5 f(x) the Maclaurin series representing 1 dt.
0 (1 + t 2 ) 2
3
y 5 4x 1 2
2
1
0 x