Week in Review # 7
MATH 142
Section 6.1, 6.2, 6.4
Drost-Spring 2010
April 4, 2010
15.
Z
e−x · x dx
16.
Z
12x2
17.
Z
x
√
dx
x + 12
18.
Z
x (x + 2)3 dx
19.
Z
x2 + 2x
dx
(x3 + 3x2 + 2)4
2
p
x3 + 1 dx
1.
Z
2.
Z
3.
Z
e dt
4.
Z
√
24 x dx
20.
Z
5.
Z
(x − ex ) dx
21.
Z
(ln x)3
dx
x
√
ln x
dx
x
6.
Z
x4 (4 + x3 ) dx
22.
Z
√
5
7.
Z
5
dt
t2
23.
Z
(−4) dx
(2x) dx
3
1 − x3
dx
x4
Z 12
3
− 2 + 1 dx
9.
x4
x
Z
1 − xex
dx
10.
x
8.
Z
11. Sales analysis. Monthly sales of a particular personal computer are expected to decline at the
rate of
2
S ′ (t) = −25t 3
computers per month, where t is time in months
and S(t) is the number of computers sold each
month. The company plans to stop manufacturing this computer when monthly sales reach 800
computers. If monthly sales now (t = 0) are 2000
computers, find S(t). How long will the company
continue to manufacture this computer? Source:
CALCULUS, by Barnett, p.371, #111
12. The rate of change of salary for a minor
√ leage
ballplayer is modeled by −400 + 1500 x over
the interval [1, 16] with S(1) = 1100. Find the
ballplayer’s salary his 10th year.
13. A bacteria culture is growing at the rate
W ′ (t) = 0.4e0.1t grams per hour. If the culture
weighed 2 grams originally, what is the weight of
the culture W (t)
ans:
x
dx
2x2 + 5
ans:
2
2
3 (ln
5
2
16 (2x
3
x) 2 + C
4
+ 5) 5 + C
2
5x e4−x dx
ans: − 25 e4−x + C
24. Find the revenue function for a purse manufacturer if the marginal revenue is given by
2
(1 − x)e2x−x , where x is the number of thou2
sands of purses sold.
ans: 21 e2x−x − 21
25. An object travels with a velocity function given
in the following figure. Find an upper and lower
estimate of the distance traveled by the object.
v
(5,25)
(4,23)
25
20
(3,20)
15
(2,12)
10
(1,5)
5
t
1
2
3
4
5
6
26. If f (x) = −(x − 4)2 + 20, approximate the area
under the curve f (x) and above the x-axis from
x = 0 to x = 6, using 3 rectangles from
a. the left.
b. the right.
27. How many rectangles should be used in the previous problem so that the error is less than 1?
a. after t hours?
b. after 90 minutes?
14.
Z
1
dx
2x + 5
28. Find the exact
Z 3 p
9 − x2 dx
0
value
using
geometry:
ans:
9
4π
29. From a study on memorizing facts, a researcher
found on average, the rate of learning facts,
N ′ (x) after x hours of studying was defined approximately by the following values:
x
N ′ (x)
0
30
2
26
4
22
6
20
8
18
10
16
12
12
Use left and right sums over three equal subintervals to approximate the area under the graph
of N ′ (x) from x = 6 to x = 12. Calculate the
error bound.
Fill in the blanks:
Z 12
≤
N ′ (x) dx ≤
6
30. Evaluate the definite integrals below given:
A = 2.4, B = 1.25, C = 0.5, D = 1.5
y
−a
A
f(x)
a
b
B
a)
b)
c)
C
c
d
x
D
d
Z
f (x) dx
−a
Z c
a
Z c
f (x) dx
f (x) dx
b
d)
Z
b
f (x) dx
d
Use the Properties of definite integrals, and
Z
3
x dx = 4.5,
Z
0
0
3
x2 dx = 9,
Z
6
x2 dx = 63
3
to evaluate each of the following:
Z
3
Z
3
32.
3
33.
Z
3
34.
Z
31.
6x dx
0
4x2 dx
0
6x2 dx
3
6
5x2 dx
35.
Z
0
3
(5x + 3x2 ) dx