Calculus Concepts And Contexts 4th Edition By James Stewart – Test Bank
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Sample Questions
Section 3.4: The Chain Rule
1. Find
the derivative of .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
None of these |
ANS:
A
PTS: 1
2. Find
the derivative of .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
None of these |
ANS:
D
PTS: 1
3. Find
the derivative of .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
None of these |
ANS:
F
PTS: 1
4. Find
the derivative of
|
a. |
0 |
e. |
16 |
|
b. |
28 |
f. |
24 |
|
c. |
4 |
g. |
12 |
|
d. |
32 |
h. |
8 |
ANS:
A
PTS: 1
5. If
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
None of these |
ANS:
C
PTS: 1
6. If
|
a. |
4 |
e. |
|
|
b. |
f. |
||
|
c. |
12 |
g. |
6 |
|
d. |
3 |
h. |
None of these |
ANS: E
PTS: 1
7. If
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
1 |
ANS:
A
PTS: 1
8. Let
Find the value of
|
a. |
2e |
e. |
|
|
b. |
e |
f. |
|
|
c. |
g. |
||
|
d. |
h. |
ANS:
E
PTS: 1
9. Let
Find the value of
|
a. |
2 |
e. |
|
|
b. |
4 |
f. |
|
|
c. |
6 |
g. |
|
|
d. |
8 |
h. |
ANS:
F
PTS: 1
10. Let
where g is
differentiable. Find
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
ANS:
E
PTS: 1
11. Suppose
that and , Find the value of
|
a. |
4 |
e. |
20 |
|
b. |
8 |
f. |
24 |
|
c. |
12 |
g. |
28 |
|
d. |
16 |
h. |
32 |
ANS:
G
PTS: 1
12. Suppose
that and , Find the value of
|
a. |
3 |
e. |
12 |
|
b. |
4 |
f. |
15 |
|
c. |
7 |
g. |
17 |
|
d. |
9 |
h. |
20 |
ANS:
E
PTS: 1
13. Suppose
that and and Find
|
a. |
5 |
e. |
25 |
|
b. |
10 |
f. |
30 |
|
c. |
15 |
g. |
35 |
|
d. |
20 |
h. |
40 |
ANS:
D
PTS: 1
14. Suppose
that , find
|
a. |
0 |
e. |
8 |
|
b. |
2 |
f. |
|
|
c. |
4 |
g. |
|
|
d. |
6 |
h. |
ANS:
D
PTS: 1
15. If
|
a. |
4 |
e. |
16 |
|
b. |
24 |
f. |
8 |
|
c. |
28 |
g. |
32 |
|
d. |
12 |
h. |
6 |
ANS: C
PTS: 1
16. If
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
1 |
g. |
|
|
d. |
h. |
ANS:
A
PTS: 1
17. If
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
ANS:
H
PTS: 1
18. If
|
a. |
0 |
e. |
|
|
b. |
2 |
f. |
64 |
|
c. |
g. |
||
|
d. |
8 |
h. |
128 |
ANS: A
PTS: 1
19. If
where k is
a constant.
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
ANS:
E
PTS: 1
20. Find
the y-intercept
of the tangent line to the curve at the point (1, 2).
|
a. |
e. |
||
|
b. |
f. |
1 |
|
|
c. |
g. |
||
|
d. |
2 |
h. |
ANS:
B
PTS: 1
21. Find
the slope of the tangent to the curve when
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
ANS:
C
PTS: 1
22. Find
the slope of the tangent to the curve , when
|
a. |
e. |
3 |
|
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
4 |
ANS:
H
PTS: 1
23. At
what value of t does
the curve have a vertical tangent?
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
||
|
d. |
h. |
ANS:
A
PTS: 1
24. Find
the slope of the tangent to the curve when .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
g. |
1 |
|
|
d. |
0 |
h. |
ANS:
G
PTS: 1
25. Given
find the value of when
|
a. |
e. |
||
|
b. |
f. |
2 |
|
|
c. |
g. |
1 |
|
|
d. |
h. |
ANS:
E
PTS: 1
26. Find
the slope of the tangent to the curve with parametric equations at the
point (0, 1).
|
a. |
e. |
1 |
|
|
b. |
f. |
2 |
|
|
c. |
g. |
3 |
|
|
d. |
0 |
h. |
4 |
ANS:
F
PTS: 1
27. Find
(a)
(b)
(c)
(d)
ANS:
(a)
(b)
(c)
(d)
PTS: 1
28. Find
(a)
(b)
(c)
(d)
ANS:
(a)
(b)
(c)
(d)
PTS: 1
29. Find
(a)
(b)
(c)
(d)
ANS:
(a)
(b)
(c)
(d)
PTS: 1
30. Find
,
(a)
(b)
(c)
(d)
ANS:
(a)
(b)
(c)
(d)
PTS: 1
31. Find
.
(a)
(b)
(c)
(d)
ANS:
(a)
(b)
(c)
(d)
PTS: 1
32. Suppose
that u and v are
differentiable functions and that and Find
ANS:
PTS: 1
33. f and g are functions
whose graphs are shown below. Let and Find each derivative, if it
exists. If it does not exist, explain.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
ANS:
(a)
(b)
(c)
(d) undefined because does not exist.
(e)
(f)
(g)
(h) undefined because does not exist.
(i)
PTS: 1
34. Suppose
that h (x) = f (g (x)) and that we are
given the following information:
Use the table to estimate the value of (0:3). Justify your
estimation.
ANS:
PTS: 1
35. Find
an equation of the tangent line to the curve at the point (1, 8).
ANS:
The slope is 24, so an equation of the tangent line is
PTS: 1
36. Find
an equation of the tangent to the curve at .
ANS:
PTS: 1
37. Find
the point where the tangent to the curve has zero slope.
ANS:
(2, 2)
PTS: 1
38. Find
the y-intercept
of the tangent line to the curve at the point (, 0).
ANS:
PTS: 1
39. According
to the theory of relativity, the mass of an object at speed v is given by
where c is
the speed of light and is the mass of the object when it is at rest. Find
.
ANS:
PTS: 1
40. The
position of a particle moving along the x-axis
is given by meters, where t is
measured in seconds.
(a) Determine the position, velocity, and acceleration of
the particle when t = 0.65.
(b) Show that the acceleration of the particle is
proportional to its position, but in the opposite direction.
ANS:
(a) ; ;
(b)
PTS: 1
41. The
angular displacement q of
a simple pendulum is given by where is the angular amplitude, w the angular
frequency and q a phase constant depending on initial conditions. If we are
given that w = 10
and , find the angular velocity when .
ANS:
; when , so . So .
PTS: 1
42. The
displacement of a particle is given by . Find all times t > 0 where
(a) The displacement attains its maximum value.
(b) The velocity attains its maximum value.
(c) The acceleration attains its maximum value.
ANS:
(a) .
(b)
(c)
PTS: 1
43. Let
be the amount of salt (in kg) in a tank after time t minutes. Find:
(a) How much salt is in the tank after 1 hour?
(b) Find the rate of change of salt after 1 hour?
ANS:
(a) .
(b)
PTS: 1
44. Let
be the population of a bacteria colony at time t hours. Find the growth rate of the bacteria
after 10 hours.
ANS:
About 2.7/h
PTS: 1
45. Let
be the population of a bacteria colony at time t hours. Find the growth rate of the
bacteria after 10 hours.
ANS:
PTS: 1
46. Consider
the two functions and
(a) Which, if either, of these functions is periodic?
Justify your answer.
(b) For each function, consider the limit as x increases without
bound. Does either function also increase without bound like an exponential
function? Explain.
(c) Where, if anywhere, does each function have an x-intercept? Justify
your answer.
(d) Where, if anywhere, does each function have a
horizontal tangent line? Justify your answer.
(e) Where, if anywhere, does each function attain its
maximum value? its minimum value? Justify your answers.
ANS:
(a) Since is periodic.
(b) No, and Both functions are bounded.
(c) are x-intercepts of g(x). for
all x,
so f has
no x-intercepts.
(d) f(x) has horizontal
tangent lines where ,
g(x)
has horizontal tangent lines where
,
(e) is maximized when that is, when ,
is maximized when , that is, when ,
is minimized when that is, when ,
is minimized when , that is, when ,
PTS: 1
47. The
function f is
graphed below.
Let and .Use the graph to estimate each of the following.
(a)
(b)
(c)
ANS:
(a)
(b)
(c) So
PTS: 1
48. Find
the derivative if where m and c are
constants, v is
velocity function.
ANS:
PTS: 1
49. Consider
the curve given by Find at the point corresponding to
ANS:
PTS: 1
50. Consider
the curve given by Find at the point corresponding to
ANS:
PTS: 1
51. Find
an equation in x and y for the tangent
line to the curve at the point
ANS:
PTS: 1
52. Find
for the parametric curve given by
ANS:
PTS: 1
53. Find
for the parametric curve given by
ANS:
PTS: 1
Section 4.2: Maximum and Minimum Values
1. Find
all critical numbers for the function .
|
a. |
0 |
e. |
8 |
|
b. |
0, 8 |
f. |
1, –1 |
|
c. |
0, 4 |
g. |
4 |
|
d. |
No Critical Number |
h. |
None of the above |
ANS:
B
PTS: 1
2. Find
all critical numbers for the function .
|
a. |
1 |
e. |
2 |
|
b. |
1, 2 |
f. |
1, –1 |
|
c. |
–1 |
g. |
–1, 2 |
|
d. |
No Critical Number |
h. |
None of the above |
ANS:
D
PTS: 1
3. Find
all critical numbers for the function .
|
a. |
0 |
e. |
3 |
|
b. |
–3 |
f. |
3, –3 |
|
c. |
0, –3 |
g. |
0, 3, –3 |
|
d. |
No Critical Number |
h. |
None of the above |
ANS:
G
PTS: 1
4. Find
all critical numbers for the function .
|
a. |
0 |
e. |
3 |
|
b. |
–3 |
f. |
3, –3 |
|
c. |
0, –3 |
g. |
0, 3, –3 |
|
d. |
No Critical Number |
h. |
None of the above |
ANS: A
PTS: 1
5. Find
all critical numbers for the function .
|
a. |
0 |
e. |
3 |
|
b. |
–3 |
f. |
3, –3 |
|
c. |
0, –3 |
g. |
0, 3, –3 |
|
d. |
No Critical Number |
h. |
None of the above |
ANS:
G
PTS: 1
6. Find
all critical numbers for the function .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
0 |
g. |
–1 |
|
d. |
None of these |
h. |
No critical numbers |
ANS:
B
PTS: 1
7. Find
all critical numbers for the function .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
0 |
g. |
–1 |
|
d. |
None of these |
h. |
No critical numbers |
ANS: F
PTS: 1
8. Find
all critical numbers for the function .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
0 |
g. |
–1 |
|
d. |
None of these |
h. |
No critical numbers |
ANS:
E
PTS: 1
9. Find
the minimum value of the function .
|
a. |
–1 |
e. |
|
|
b. |
f. |
0 |
|
|
c. |
g. |
||
|
d. |
1 |
h. |
ANS:
D
PTS: 1
10. Find
the value x at
which the minimum of the function occurs.
|
a. |
1 |
e. |
0 |
|
b. |
f. |
||
|
c. |
g. |
–1 |
|
|
d. |
h. |
ANS:
A
PTS: 1
11. Find
the distance between the two critical numbers of the function .
|
a. |
4 |
e. |
2 |
|
b. |
1 |
f. |
9 |
|
c. |
8 |
g. |
6 |
|
d. |
3 |
h. |
5 |
ANS:
E
PTS: 1
12. Find
the difference between the local maximum and the local minimum values of the
function .
|
a. |
4 |
e. |
6 |
|
b. |
1 |
f. |
5 |
|
c. |
9 |
g. |
8 |
|
d. |
2 |
h. |
3 |
ANS:
A
PTS: 1
13. Find
the absolute maximum of the function on the interval .
|
a. |
e. |
1 |
|
|
b. |
f. |
||
|
c. |
0 |
g. |
|
|
d. |
h. |
2 |
ANS:
H
PTS: 1
14. Find
the absolute minimum and maximum values of the function on the closed
interval .
|
a. |
0, 3 |
e. |
3, 16 |
|
b. |
0, 5 |
f. |
5, 7 |
|
c. |
3, 5 |
g. |
7, 16 |
|
d. |
3, 9.75 |
h. |
5, 10.25 |
ANS:
E
PTS: 1
15. Find
the minimum and maximum values of on the interval .
|
a. |
e. |
||
|
b. |
f. |
||
|
c. |
0, 8 |
g. |
|
|
d. |
h. |
ANS: B
PTS: 1
16. Given
that has critical numbers at find a and b.
|
a. |
6 |
e. |
9, 3 |
|
b. |
8, 7 |
f. |
8, 4 |
|
c. |
7, 8 |
g. |
7, 5 |
|
d. |
6, 9 |
h. |
6, 6 |
ANS:
D
PTS: 1
17. Find
the absolute maximum of the function
|
a. |
e. |
2 |
|
|
b. |
1 |
f. |
|
|
c. |
g. |
||
|
d. |
h. |
No absolute minimum |
ANS:
H
PTS: 1
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