AP Calculus (BC) Chapter 5 TestNo Calculator Section

Name: Date: Period:

Part I. Multiple-Choice Questions (5 points each; please circle the correctanswer.)

1. The graph of the function f shown below consists of a semicircle and

a straight line segment. Then∫ 4

−2f(x) dx =.

(A) 2π + 1

(B) 2π − 1

(C) π + 2

(D) π − 2

(E) π − 1

2.∫

ln 2x dx =

(A)ln 2x

x+ C

(B)ln 2x

2x+ C

(C) x lnx− x+ C

(D) x ln 2x− x+ C

(E) 2x ln 2x− 2x+ C

3.d

dx

∫ x2

−2

t dt

1 +√t

=

(A)2x

1 + x

(B)x2

1 + |x|

(C)2x3

1 + |x|

(D)2t3

1 + | t|

(E)2x3

1 + x

4. If∫ 20

−10f(x) dx = A and

∫ 20

10f(x) dx = B, then

∫ 10

−10f(x) dx =

(A) A+B

(B) A−B(C) 0(D) B − A(E) 20

5.∫ ∞

0e−x dx =

(A) 0(B)∞(C) −∞(D) −1

(E) 1

Part II. Free-Response Questions

6. Assume that a water pump is pumping water into a large tank at a

variable rate: after t hours, water is being pumped at a rate of v(t) =500t

1 + tgallons/hour.

(a) (5 points) Write down the definite integral that expresses how muchwater has been pumped into the vessel after 24 hours. (Don’t com-pute this.)

(b) (7 points) Compute how much water is in the tank after 24 hours,assuming that the tank was empty to begin with and that F (t) =500 (t− ln(1 + t)) is an antiderivative for v(t).

7. You are given a function f whosederivative f ′ has been graphed onthe closed interval −3 ≤ x ≤ 4.You are given that f(0) = π.(a) (6 points) Find f(−3) and f(2).

-3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

x

y

(-3,1)

(4,-2)

Graph of f'.

.

(b) (3 points) Find the average value of f ′ on the interval [−3, 4].

(c) (4 points) Find the maximum value of f on the interval [−3, 4]. Jus-tify your reasoning.

AP Calculus (BC) Chapter 5 TestCalculators Allowed

Name: Date: Period:

Part III. Multiple-Choice Questions (5 points each; please circle thecorrect answer.)

8. The average value of the function f(x) = ln2 x on the interval [2, 4] is

(A) −1.204

(B) 1.204

(C) 2.159

(D) 2.408

(E) 8.636

9. Using the Trapezoid Rule with four trapezoids of equal width to ap-proximate the area between the curve y = x3 − x2 and the x-axis overthe interval [3, 4] gives the approximation

(A) 35.266

(B) 27.766

(C) 63.061

(D) 31.516

(E) 25.125

10. A particle is moving with velocity v(t) = 7e−t2, where t stands for time.The total distance traveled (to three decimal places) by this particle inthe first four seconds is

(A) 0.976

(B) 6.204

(C) 6.359

(D) 12.720

(E) 7.000

11. The graph of the function f ,consisting of three line seg-ments, is shown. Let

g(x) =

∫ x

−2f(t) dt.

Then g′′(0) =

(A)4

3(B) 2.6

(C) undefined(D) 0

(E) −3

12.∫x√

3x dx =

(A)2√

3

5x

52 + C

(B)5√

3

2x

52 + C

(C)5√

3

2x

12 + C

(D) 2√

3x+ C

(E)5√

3

2x

32 + C

Part IV. Free-Response Questions

13. Traffic flow is defined as the rate at which cars pass through an inter-section, measured in cars per minute. The traffic flow at a particularintersection is modeled by the function F defined by

F (t) = 82 + 4 sin

(t

2

)for 0 ≤ t ≤ 30,

where F (t) is measured in cars per minute and t is measured in min-utes.

(a) (6 points) To the nearest whole number, how many cars pass throughthe intersection over the 30-minute period?

(b) (6 points) What is the average rate of change of the traffic flowover the first 15 minutes? Indicate units of measure.

x

y

y=f(x)

y=18-3x

RS

14. Let f be the function given by f(x) = 4x2 − x3, and let L be the liney = 18 − 3x, where L is tangent to the graph of f . Let R be the re-gion bounded by the graph of f and the x-axis, and let S be the regionbounded by the graph of f , the line l, and the x-axis, as shown above.

(a) (5 points) Show that L is tangent to the graph of y = f(x) at thepoint x = 3.

(b) (8 points) Find the area of S.