54
Chapter 2
Differentiation
Test Form A
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
1. If f x 2x2 4, which of the following will calculate the derivative of f x?
(a)
2x x2 4 2x2 4
x
2x2 4 x 2x2 4
x→0
x
(b) lim
2x x2 4 2x2 4
x→0
x
(c) lim
2x2 4 x 2x2 4
x
(e) None of these
(d)
2. Differentiate: y (a) 1
(d)
1 cos x
.
1 cos x
2 sin x
1 cos x2
(b) 2 csc x
(c) 2 csc x
(e) None of these
3. Find dydx for y x 3x 1.
3x2
2x 1
(b)
(d)
7x3 x2
2x 1
(e) None of these
(c) 3x2x 1
4. Find fx for f x 2x2 57.
(a) 74x6
(b) 4x7
(d) 72x2 56
(e) None of these
5. Find
(c) 28x2x2 56
d 2y
for y cos2 4x.
dx2
(a) 8 cos 4x
(b) 32 sin 4x
(d) 32 cos 8x
(e) None of these
(c) 4 cos 8x
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x27x 6
2x 1
(a)
Chapter 2
Test Bank
6. The position equation for the movement of a particle is given by s t2 13 when s is measured
in feet and t is measured in seconds. Find the acceleration at two seconds.
(a) 342 unitssec2
(b) 18 unitssec2
(d) 90 unitssec2
(e) None of these
7. Find
(c) 288 unitssec2
dy
if y2 3xy x2 7.
dx
(a)
2x y
3x 2y
(b)
3y 2x
2y 3x
(d)
2x
y
(e) None of these
(c)
2x
3 2y
8. Find y if y sinx y.
cosx y
1 cosx y
(a) 0
(b)
(d) 1
(e) None of these
(c) cosx y
9. Differentiate: y sec2 x tan2 x.
(a) 0
(b) tan x sec4 x
(d) 4 sec2 x tan x
(e) None of these
(c) sec2 x sec2 x tan2 x
t
10. Find the derivative: s t csc .
2
(a) csc
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(d)
t
t
cot
2
2
1 2t
cot
2
2
1
t
(b) cot2
2
2
(c)
t
t
1
csc cot
2
2
2
(e) None of these
11. Find an equation for the tangent line to the graph of f x 2x2 2x 3 at the point where x 1.
(a) y 2x 2
(b) y 4x2 6x 5
(d) y 4x2 6x 2
(e) None of these
(c) y 2x 1
12. Find all points on the graph of f x x3 3x2 2 at which there is a horizontal tangent line.
(a) 0, 2, 2, 2
(b) 0, 2
(d) 2, 2
(e) None of these
(c) 1, 0, 0, 2
55
56
Chapter 2
Differentiation
13. Find the instantaneous rate of change of w with respect to z if w (a)
7
6z
(d) (b)
14
3z3
7
.
3z2
14
z
3
(c) 14
3z
(e) None of these
14. Let px f xgx. Use the figure to find p5.
(a) 7
y
8
(b) 3
f
6
(c) 0
(d) 24
g
(e) None of these
2
x
−2
2
4
6
8
−2
15. A point moves along the curve y 2x2 1 in such a way that the y value is decreasing at the rate
of 2 units per second. At what rate is x changing when x 32?
(a) increasing 13 unitsec
(b) decreasing 13 unitsec
(d) increasing 72 unitsec
(e) None of these
(c) decreasing 72 unitsec
16. Assume fc 4. Find f c if f is an odd function.
(b) 0
(c) 3
(d) 4
(e) None of these
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(a) 4
Chapter 2
57
Test Form B
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
1. If f x x2 x, which of the following will calculate the derivative of f x?
x2 x x x2 x
x→0
x
(a) lim
x x2 x x x2 x
x→0
x
(b) lim
(c)
x x2 x x x2 x
x
x2 x x x2 x
x
(e) None of these
(d)
2. Differentiate: y 3x
.
x2 1
(a)
3
1 x2
(b)
(d)
31 x2
1 x22
(e) None of these
3. Find
(a)
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Test Bank
(d)
3
2x
(c)
3x2 3
1 x23
dy
for y x 3x 1.
dx
9x 1
2x
3
2x
(b)
9
x 1
2
(c) 3x
(e) None of these
4. Find fx for f x sin 4x.
(a)
(d)
5. Find
2
sin 4x
2 cos 4x
sin 4x
(b) sin 8x
(c)
cos 4x
2sin 4x
(e) None of these
d 2y
for y 3x2 13.
dx 2
(a) 183x2 115x2 1
(b) 63x2 1
(d) 216x3x2 1
(e) None of these
(c) 18x3x2 12
58
Chapter 2
6. Find
(a)
Differentiation
dy
if x2 y2 2xy.
dx
x
1y
(d) x
y
(b)
yx
yx
(c) 1
(e) None of these
7. Find y if x tanx y.
(a) sin2 x y
(d)
1 sec2 x
sec2 y
(b) sec2 x y
(c) tan2 x y
(e) None of these
8. Differentiate: y csc2 cot2 .
(a) cot csc4 (b) 0
(c) 4 csc2 cot (d) csc2 csc2 cot2 (e) None of these
9. Find the derivative: s t sec t.
(a) tan2t
(d) sect tant
sect tant
2t
(e) None of these
(b)
(c) sec
1
1
tan
2t
2t
10. The position equation for the movement of a particle is given by st t 3 1 where s is measured in feet and
t in measured in seconds. Find the acceleration of the particle at 2 seconds.
2
3
(a) 3 feetsec2
(b)
(d) 19 footsec2
(e) None of these
footsec2
1
(c) 108
footsec2
(a) y 4x 2
(b) 2x y 1 0
(d) 2x y 5
(e) None of these
(c) y 4x2 2x 1
12. Find point(s) on the graph of the function f x x3 2 where the slope is 3.
(a) 1, 3, 1, 3
(b) 1, 1, 1, 3
(d) 1, 3
(e) None of these
13. Find the instantaneous rate of change of w with respect to z for w (a)
3
2
(b) 2
(d)
1
z2
(e) None of these
(c)
3 2, 0
z
1
.
z
2
(c)
z2 2
2z2
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11. Find an equation for the tangent line to the graph of f x 2x2 2x 3 at the point where x 1.
Chapter 2
14. Let qx f x
. Use the figure to find q5.
gx
y
8
3
2
(a) 3
(b)
3
(c) 16
3
(d) 4
Test Bank
f
6
g
2
(e) None of these
x
−2
2
4
6
8
−2
15. A point moves along the curve y 2x2 1 in such a way that the y value is decreasing at the rate of
3
2 units per second. At what rate is x changing when x 2?
7
(a) decreasing 2 unitsec
7
(b) increasing 2 unitsec
1
(d) decreasing 3 unitsec
(e) None of these
1
(c) increasing 3 unitsec
16. Assume fc 4. Find fc if f is an even function.
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(a) 4
(b) 0
(c) 8
(d) 4
(e) None of these
59
60
Chapter 2
Differentiation
Test Form C
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator is needed for some problems.
1. Determine whether the slope at the indicated point is positive, negative, or zero.
y
3
(a) Zero
2
(b) No slope
1
(c) Positive
−3
x
−2
1
2
3
−1
(d) Negative
−2
−3
(e) None of these
2. Find the slope of the graph of f x x2 2x at the point a, f (a).
(c) f a
(a) 0
(b) 2a 2
(d) a2 2a
(e) None of these
3. Use a graphing calculator to find the x-values at which f is not differentiable for f x 2x 645.
(a) 3
(b) 0
(c) f is differentiable everywhere.
(d) 3
(e) None of these
3x2 8
and its derivative, f, on the same coordinate axes.
x2 4
Then use the graph to describe the behavior of f at that value of x where fx 0.
4. Use a graphing calculator to graph f x (b) f increases without bound.
(c) f has a horizontal tangent line.
(d) f has no tangent line.
(e) None of these
5. Find the value of the derivative of the function f t t3 2
at the point 2, 3.
t
(a) 9
2
(b) 7
2
(d) 11
16
(e) None of these
(c) 12
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(a) f x 0
Chapter 2
Test Bank
6. The graph at the right represents the graph of the derivative of which of the following functions?
(a) f x 2x2 1
y
(b) f x 2x 3
3
2
(c) f x 3x2 2x 1
1
(d) f x x x
3
2
−3
−2
1
−1
(e) None of these
7. Find
x
−1
−2
dy
: y 4 sin y 5 cos x x.
dx
(a)
5 sin x
1 4 cos y
(b) 4 cos y 5 sin x 1
(d)
5 sin x
4 cos y
(e) None of these
(c)
5 sin x
4 cos y
2
8. The position function for a particular object is s 35
2 t 58t 91. Which statement is true?
(a) The initial velocity is 35.
(b) The velocity is a constant.
(c) The velocity at time t 1 is 23.
(d) The initial position is 35
2.
(e) None of these
9. Find
dy
for y csc cot .
d
(a) 0
(b) cot2 csc cot (d) csc cot csc2 (e) None of these
(c) sec tan sec2 © Houghton Mifflin Company. All rights reserved.
10. Find an equation of the tangent line to the graph of f tan at the point
(a) 4x 4y 4
(b) 42x 4y 4
(d) y x
(e) None of these
4 , 1.
(c) 4x 2y 2
1
f x
11. Let f 3 0, f3 6, g3 1 and g3 . Find h3 if hx .
3
gx
(a) 18
(b) 6
(d) 2
(e) None of these
12. Find the derivative: f x 1
3 3 x3
(c) 6
.
(a)
1
33 x343
(b)
(d)
x2
3 x343
(e) None of these
x2
3 x343
(c)
x2
3 x323
2
3
61
62
Chapter 2
Differentiation
13. Find the derivative: f sin 2.
cos 2
sin 2
(d) cos (a)
(b) sec 2
(c)
cos 2
2sin 2
(e) None of these
14. Determine the slope of the graph of the relation 2x2 3xy y3 1 at the point 2, 3.
17
(a) 21
(d)
4
3
(b)
1
(c) 3
5
7
(e) None of these
15. A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at a
constant rate of 0.05 inches per second and the volume V is 128 cubic inches. At what rate is
the length h changing when the radius r is 1.8 inches? Hint: V r2h
(a) 2.195 in.sec
(b) 39.51 in.sec
(d) 43.90 in.sec
(e) None of these
(c) 2.195 in.sec
16. Let px f xgx. Use the figure to find p8.
(a) 3
(b) 1
(c) 28
(d) 11
y
8
f
6
g
(e) None of these
2
x
−2
2
4
6
8
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−2
Chapter 2
Test Bank
Test Form D
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
1
1. Use the definition of a derivative to find the derivative of f x .
x
2. Differentiate: y 3. Find
2x
.
1 3x2
dy
for y x32x 1.
dx
4. Find fx for f x cot3x.
5. Calculate
d 2y
1x
for y .
dx2
2x
6. The position equation for the movement of a particle is given by s t3 12 where s is measured
in feet and t is measured in seconds. Find the acceleration of this particle at one second.
7. Find y if y 8. Find
x
.
xy
dy
if x cos y.
dx
9. Find the derivative: f sec 2.
10. Differentiate and simplify: y sin2 x cos2 x.
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63
11. Find an equation for the tangent line to the graph of f x x 1 at the point where x 3.
12. Find the values of x for all points on the graph of f x x3 2x2 5x 16 at which the slope
of the tangent line is 4.
1
13. Find the instantaneous rate of change of R with respect to x if R 2x2 .
x
14. An object is thrown (straight down) from the top of a 220-foot building with an initial velocity of
26 feet per second.
a. Write the position equation for the movement described.
b. What is the velocity at one second?
64
Chapter 2
Differentiation
15. As a balloon in the shape of a sphere is being blown up, the volume is increasing at the rate of
4 cubic inches per second. At what rate is the radius increasing when the radius is 1 inch?
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16. Analytically show that the graph of the function f x x 3 2x 2 6x does not have a tangent line with
a slope of 4.
Chapter 2
Test Bank
Test Form E
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator is needed for some problems.
1. At each point indicated on the graph, determine whether
the value of the derivative is positive, negative, zero, or
the function has no derivative.
y
a
b c
d
e
x
4
2. Let f x .
x
a. Use the definition of the derivative to calculate the derivative of f.
b. Find the slope of the tangent line to the graph of f at the point 2, 2.
c. Write an equation of the tangent line in part b.
d. Use a graphing calculator to graph f and the tangent line on the same axes. Then sketch the graphs.
3. Find the point(s) on the graph of y 1
where the graph is parallel to the line 4x 9y 3.
x
4. A coin is dropped from a height of 750 feet. The height, s, (measured in feet), at time, t
(measured in seconds), is given by s 16t2 750.
a. Find the average velocity on the interval 1, 3.
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65
b. Find the instantaneous velocity when t 3.
c. How long does it take for the coin to hit the ground?
d. Find the velocity of the coin when it hits the ground.
3
r . How fast is the
3
volume changing with respect to changes in r when the radius is equal to 2 feet?
5. The volume of a right circular cone of radius r and height r is given by V 6. Find the derivative: f x 5 sec x tan x.
7. Evaluate the derivative for the function f t t
2
at the point ,
.
cos t
3 3
66
Chapter 2
Differentiation
8. Let f x x5 5x.
a. Calculate f x.
b. Use a graphing calculator to graph f and f on the same axes. Sketch the graphs.
c. Use the graph to determine those point(s) where f has a horizontal tangent line.
d. Give the value of f at each of the points found in part c.
9. The graphs of a function f and its derivative f are given on the same
coordinate axes. Label the graphs as f or f and state the reasons for
your choice.
y
3
2
1
−3
x
2
3
10. Find f x: f x 2x2 5.
11. Find the point(s) (if any) of horizontal tangent lines: x2 xy y2 6.
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12. A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer.
Find the rate of change of the angle of elevation when the balloon is 25 feet above the ground.