5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 305

13.7 Cumulative Review Problems

(Calculator) indicates that calculators are
permitted.

21. If

∫a

−a

ex^1 dx=k, find

∫a

0

ex^2 dxin terms

ofk.

22. A man wishes to pull a log over a
    9-foot-high garden wall as shown in
    Figure 13.7-1. He is pulling at a rate of
    2 ft/sec. At what rate is the angle between
    the rope and the ground changing when
    there are 15 feet of rope between the top of
    the wall and the log?

θ

9 ft

wall

rope

rope

log

Figure 13.7-1

23. (Calculator) Find a point on the parabola
    y=

1

2

x^2 that is closest to the point (4, 1).

24. The velocity function of a particle moving
    along thex-axis isv(t)=tcos(t^2 +1)
    fort≥0.

(a) If att=0, the particle is at the origin,

find the position of the particle att=2.

(b) Is the particle moving to the right or

left att=2?

(c) Find the acceleration of the particle

att=2 and determine if the velocity of

the particle is increasing or decreasing.

Explain why.

25. (Calculator) Given f(x)=xexand
    g(x)=cosx, find:

(a) The area of the region in the first

quadrant bounded by the graphs

f(x),g(x), andx=0.

(b) The volume obtained by revolving the

region in part (a) about thex-axis.

13.8 Solutions to Practice Problems

Part A The use of a calculator is not

allowed.

1. (a) F(0)=

∫ 0

0

f(t)dt= 0

F(3)=

∫ 3

0

f(t)dt

=

1

2

( 3 + 2 )( 4 )= 10

F(5)=

∫ 5

0

f(t)dt

=

∫ 3

0

f(t)dt+

∫ 5

3

f(t)dt

= 10 +(−4)= 6

(b) Since

∫ 5

3

f(t)dt≤0,Fis decreasing

on the interval [3, 5].

(c) Att=3,Fhas a maximum value.

(d)F′(x)=f(x),F′(x) is increasing on

(4, 5) which impliesF≤(x)>0.

Thus,Fis concave upward on (4, 5).