Exam
Name___________________________________
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
1)
Use implicit differentiation to determine dy
dx where x2+y2= 4. Is dy
dx =x
y correct?
Enter “yes” or “no”.
1)
2)
The radius, r, of a sphere is increasing. For what value of r is dV
dt equal to 64 times the
rate of increase of r. Enter just an integer.
2)
3)
Find all x–coordinates of points (x, y) on the graph of Q(x) =x3
(x + 2)5 where the tangent
line is horizontal. Enter your answer as just a, or a, b where these are integers and a< b.
3)
4)
Find the equation of the line tangent to the graph of y = x(x – 5)4 at the point (3, 48).
Enter your answer in standard point–slope form.
4)
5)
The radius of a spherical balloon increases at a rate of 1 mm/sec. How fast is the surface
area increasing when the radius is 10mm? (Note: The surface area S of a sphere of radius r
is S = 4r2.) Enter just a real number (no approximations, no units or words).
5)
6)
Find the slope of the tangent line to f(x) = 4(x3+ 1)(2x2+ 2x + 1)4 at (–1, f(–1)).
Enter just an integer.
6)
Differentiate.
7)
f(x) = (x3– 2)3(x3+ 2)4 at x = – 1
Enter just an integer.
7)
1
8)
Use implicit differentiation to determine the slope of the graph of x1/2 +y1/2 = 4 at (1, 9).
Enter just an integer.
8)
9)
Suppose that the cost of manufacturing x units of a product is C(x) = 6x – 2 x+ 1 dollars
and that the production level t weeks from the present is x = 4t2. Find the rate of change in
cost with respect to time. Enter your answer as a polynomial in t in standard form (not
labeled).
9)
10)
Compute dy
dx using the chain rule where y = 4u2+ 8u + 4 and u = 3x + 1.
Enter your answer as just a polynomial in x in standard form (no label).
11)
Determine the rate of change of 2x + 4 with respect to x at x = 1.
Enter just a reduced quotient of form a
b.
12)
Compute dy
dx using the chain rule where y =u2 and u = 3x + 4.
Enter your answer as just a polynomial in x in standard form (no label).
Differentiate.
13)
f(x) = (4x2+ 4)(2x2+ 2x) at x = 1
Enter just an integer.
14)
f(x) =x5·x – 1
x + 1 at x = 1
Enter just a reduced fraction.
15)
f(x) =x2
x + 1 at x = 2
Enter just a reduced fraction.
16)
f(x) = (x2+ 1)3(x3– 1)2 at x = – 1
Enter just an integer.
17)
When a manufacturer produces and sells x units per week, its weekly profit is P dollars,
where P = 100(2000 + 120x –x2). Production level t weeks from the present will be
x =t
2– 16. Find the rate of change in profit with respect to x. Enter your answer as a
polynomial in x in standard form (not labeled).
18)
Find the slope of the tangent line to f(x) =1
x2+ 1 at (–1, f(–1)).
Enter a reduced fraction.
19)
Find the slope of the tangent line to the graph of y =x
(8 –x2)
at the point (2, 1).
Enter just an integer.
20)
Find the equation of the line tangent to the graph of x2y3= 1 at the point (1, 1).
Enter your answer in slope–intercept form.
21)
Use implicit differentiation to determine dy
dx where x2y3+ x = y. Is dy
dx =2xy3+ 1
–3x2y2+1
correct?
Enter “yes” or “no”.
22)
Use the chain rule to compute the derivative of (3x + 1)3
(3x + 1)3+ 1 at x = – 1.
Enter just a reduced fraction.
23)
Compute dy
dx using the chain rule where y =u2
u2+ 1 and u =2x + 4 at x = 1.
Enter just a reduced fraction.
Differentiate.
24)
f(x) = (x5– 8x2+ 5)(x3+x– 1) at x = 1
Enter just an integer.
25)
f(x) = (x2+ x)(3x2+ 4x – 1) at x = 1
Enter just an integer.
26)
Find the equation of the line tangent to the graph of x2(y – 1)3= 8 at the point (1, 3).
Enter your answer in slope–intercept form.
27)
Let f(x) =x2– x
x, g(x) =1
x . Compute f(g(x)) at x= 4.
Enter just a reduced fraction.
28)
Use the chain rule to find the derivative of 6x3– 2x at x = 1.
Enter just an integer.
29)
Find the equation of the tangent line to the graph of x2+y2= 1 at the point 1
2, 3
2.
Is y = – 1
3x +2 3
3 correct?
Enter “yes” or “no”.
30)
Use implicit differentiation to determine dy
dx where x3+y3= 2xy. Is dy
dx =2y – 3x2
3y2– 2x
correct?
Enter “yes” or “no”.
31)
Use implicit differentiation to determine dy
dx where 4x3+ 4xy + y = 8. Is dy
dx =
–12x2– 4y
4x + 1
correct?
Enter “yes” or “no”.
Differentiate.
32)
f(x) =6x – 7
x2+ 5 at x = – 1
Enter just a reduced fraction.
33)
Find the slope of the tangent line to f(x) =x6+ 4x3+ 1
x3+ 1 at (1, f(1)).
Enter just a reduced fraction of form a
b.
34)
The cost of manufacturing x units is C dollars, where C = 4x + 6 x+ 5. Weekly production
at t weeks from the present is estimated to be x = 2800 + 100t units when t = 8. Find the
time rate of change of cost, dC
dt . Enter just an integer.
5
Differentiate.
35)
f(x) = (2x + 1)(3x – 2)2 at x = 1
Enter just an integer.
36)
Differentiate: f(x) = ((x5+ 2)3+ 1)4 at x = – 1.
Enter just an integer.
37)
Find the slope of the tangent line to f(x) =x
x2+ 1 at (2, f(2)).
Enter just a reduced fraction.
38)
Mr. Smith is 6 ft tall and walks at a constant rate of 2 ft/sec toward a street light that is 10 ft
above the ground. At what rate is the length of his shadow changing when he is 6 ft from
the base of the pole that supports the light? Enter just an integer (no units).
Differentiate.
39)
f(x) =x2
x3– 5x + 2 at x = – 1
Enter a reduced fraction only.
40)
Use implicit differentiation to determine the slope of the graph of x + y = x at (2, 2).
Enter just an integer.
41)
Use implicit differentiation to determine dy
dx where xy + 10 =x2. Is dy
dx =2x – y
x correct?
Enter “yes” or “no”.
42)
Find the slope of the tangent line to f(x) =x2– 1
x + 2 at (–1, f(–1)).
Enter just an integer.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Differentiate.
43)
g(x) = (x3+ 1)(3x2– 1)
43)
A)
3x4– 3x2– 6x
B)
15x4– 3x2+ 6x
C)
–3x4– 3x2+ 6x
D)
–3x4+ 3x2+ 6x
Solve the problem.
44)
A ladder is slipping down a vertical wall. If the ladder is 20 ft long and the top of it is slipping at
the constant rate of 5 ft/s, how fast is the bottom of the ladder moving along the ground when the
bottom is 16 ft from the wall?
44)
A)
3.8 ft/s
B)
0.31 ft/s
C)
0.8 ft/s
D)
6.3 ft/s
Differentiate.
45)
f(x) =1 –8x
45)
A)
–8
1 –8x
B)
–4
1 –8x
C)
1
21 –8x
D)
–4x
1 –8x
Compute f(g(x)) for the given f(x) and g(x).
46)
f(x) =6
x – 7 and g(x) =5
7x
46)
A)
5x – 35
42x
B)
42x
5+ 49x
C)
6x
5– 49x
D)
42x
5– 49x
Find d2y
dx2.
47)
y =(x2+ 1)10
47)
A)
(380x – 20)(x2+ 1)8
B)
(380x2+ 20)(x2+ 1)8
C)
(360x2– 20)(x2+ 1)8
D)
(360x2+ 20)(x2+ 1)9
Differentiate.
48)
y =x+6
x–6
48)
A)
6
2 x ( x –6) 2
B)
–x+6
( x –6) 2
C)
–6
x ( x –6) 2
D)
–6 x
( x –6) 2
49)
The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x).
1
x2+ 2x – 3
49)
A)
f(x) = x
g(x) =x2+ 2x – 3
B)
f(x) =1
x
g(x) =x2+ 2x – 3
C)
f(x) =x2+ 2x – 3
g(x) =1
x
D)
f(x) =1
x
g(x) =1
x2+ 2x – 3
Differentiate implicitly to find the slope of the curve at the given point.
50)
y3+ yx2+x2– 3y2= 0; (1, 1)
50)
A)
2
B)
–1
2
C)
–1
D)
3
2
51)
If y =u and u =x3– 5x2+ 1, find dy
dx .
51)
A)
x3–5x2+ 1
B)
3x2– 10x (x3–5x2+ 1)
C)
3x2– 10x
D)
3x2– 10x
2 x3– 5x2+ 1
52)
Suppose 2x3– 3p4= 6, where x and p are differentiable functions of t. Find dp
dt .
52)
A)
6 – 6x2
12p3
dx
dt
B)
6x2dx
dt – 12p3
C)
x2
2p3
dx
dt
D)
6x2– 12p3dp
dt
53)
The following function may be viewed as a composite function f(g(x)). Find f(x) and g(x).
(3x2– 2x + 1)6
53)
A)
f(x) = 3x2– 2x + 1
g(x) =x6
B)
f(x) =x6
g(x) = 3x2– 2x + 1
C)
f(x) =x6
g(x) =(3x2– 2x + 1)6
D)
f(x) = x
g(x) = 3x2– 2x + 1
Solve the problem.
54)
Electrical systems are governed by Ohm’s law, which states the
V = IR, where V = voltage, I = current, and R = resistance. If the current in an electrical system is
decreasing at a rate of 5 A/s while the voltage remains constant at 20 V, at what rate is the resistance
increasing when the current is 48 A?
54)
A)
25
576 ohms/s
B)
125
12 ohms/s
C)
12
25 ohms/s
D)
25
12 ohms/s
Differentiate.
55)
f(x) = (4x – 4)(3x3– x2+ 1)
55)
A)
48x3– 48x2+ 8x + 4
B)
48x3– 16x2+ 48x + 4
C)
36x3+ 48x2– 16x + 4
D)
12x3+ 16x2– 48x + 4
56)
f(x) = (3x – 6)(6x + 1)
56)
A)
36x – 16.5
B)
36x – 33
C)
36x – 39
D)
18x – 33
Compute f(g(x)) for the given f(x) and g(x).
57)
f(x) =5x –6 and g(x) =x5+6
57)
A)
x
B)
x5
C)
–x
D)
1
58)
If y = 3u + 2 and u =t
t + 1 , find dy
dt .
58)
A)
1
(t + 1)2
B)
1
9
C)
1
(3u + 2)2
D)
3
(t + 1)2
10
Differentiate.
59)
y =x + 1
x2– 1
59)
A)
x3+x2– x – 1
(x2– 1)2
B)
–1
(x – 1)2
C)
1
2x
D)
none of these
60)
f(x) =1
(4x2– 5x – 5)3
60)
A)
–3(8x – 5)
(4x2– 5x – 5)4
B)
–3
(4x2– 5x – 5)4
C)
(8x – 5)
(4x2– 5x – 5)4
D)
–3(8x – 5)
(4x2– 5x – 5)3
61)
If y = ( 3x+ 1)5, find dy
dx .
61)
A)
1
3(3x+ 1)4x–2/3
B)
5
3(3x+ 1)4x–2/3
C)
5( 3x+ 1)4
D)
5( 3x+ 1)43x
E)
none of these
11
Find dy/dx by implicit differentiation.
62)
y2– xy +x2=4
62)
A)
2x – y
x + 2y
B)
2x + y
x + 2y
C)
2x – y
x – 2y
D)
2x + y
x – 2y
Solve the problem.
63)
Given the revenue and cost functions R(x) =32x – 0.4x2 and C(x) =6x + 13, where x is the daily
production, find the rate of change of profit with respect to time when x =15 units and
dx
dt =8 units per day.
63)
A)
$201.6/day
B)
$196/day
C)
$160/day
D)
$112/day
64)
The demand function for a certain product is given by:
D(p) =7p +280
10p +13 .
Find the marginal demand D'(p).
64)
A)
D'(p) =
–2709
(10p +13)2
B)
D'(p) =
–2709
10p +13
C)
D'(p) =2891 +140p
(10p +13)2
D)
D'(p) =2709
(10p +13)2
65)
The total cost to produce x units of perfume is C(x) = (7x + 6)(9x + 8). Find the marginal average
cost function.
65)
A)
126 –110
x
B)
63x + 110 +48
x
C)
126x + 110
D)
63 –48
x2
12
66)
Suppose that x and y are related by the equation x2
4+y3
2= 4. Use implicit differentiation to
determine dy
dx .
66)
A)
dy
dx =2y2
x
B)
dy
dx = – x
3y2, y
0
C)
dy
dx = – 2x
y3
D)
dy
dx =1
3y2, y 0
Differentiate.
67)
y = (5–2x2)(6x2–48)
67)
A)
12x3+126x
B)
–48x3+252
C)
–48x4+252x2
D)
–48x3+252x
68)
f(t) =t3
t–3
68)
A)
5t3–18t2
2 t ( t –3)2
B)
t5/2 –9t2
2( t –3)2
C)
5t5/2 –18t2
2( t –3)2
D)
5t3–18t2
2( t –3)2
Express the given function H as a composition of two functions f(x) and g(x) such that H(x) = f(g(x)).
69)
H(x) =1
x2– 9
69)
A)
f(x) =1
x2, g(x) = – 1
9
B)
f(x) =1
9, g(x) = x2– 9
C)
f(x) =1
x2, g(x) = x – 9
D)
f(x) =1
x, g(x) = x2– 9
70)
Let f(x) =x3. Using the chain rule, find an expression for the derivative of [f(g(x))].
70)
A)
3x2 g'(x)3
B)
3[g(x)]2
C)
3[g(x)]2 g'(x)
D)
g'(x)3
E)
none of these
71)
If f(x) = x(x – 1)5 and g(x) =x2, find d
dx f(g(x)).
71)
A)
(x2– 1)5+ 5x2(x2– 1)4
B)
2x(x2– 1)5+ 10x3(x2– 1)4
C)
2x(x2– 1)5+ 10x2(x2– 1)4
D)
2x(x2– 1)5+ 10x3(x2– 1)
72)
One hour after x milligrams of a particular drug are given to a person, the change in body
temperature T(x), in degrees Celsius, is given approximately by: T(x) =5x2
91 –x
9–160
9, 0
x
6.
Find the sensitivity, T'(x), of the body to a dosage of three milligrams.
72)
A)
–5
3 degrees per milligram
B)
–10
27 degree per milligram
C)
–10
9 degrees per milligram
D)
5
3 degrees per milligram
Differentiate.
73)
f(x) =2x – 7
3x – 2
73)
A)
–17
(2x – 7)2
B)
–17
(3x – 2)2
C)
17
(3x – 2)2
D)
17
(2x – 7)2
74)
y =4x(4x3–7x)
74)
A)
48x3–28x
B)
64x3–28x
C)
64x3–56x
D)
48x3–56x
75)
If f(x) =x2– 9 and g(x) =x2– 16, find d
dx g(f(x)).
75)
A)
4x3– 36x
B)
(x2– 9)2– 16
C)
((x2– 9)2– 16)(2x)
D)
4x3– 50x2
D)
Find dy/dx by implicit differentiation.
76)
5y2+ 3x2=3
76)
A)
–6x +3
10y
B)
–3x2
10y
C)
–3x
5y
D)
–3x
5
D)
Differentiate.
77)
q(t) =5t
t2–3t – 6
77)
A)
–5(t2–3t + 6)
(t2–3t – 6)2
B)
5
2t –3
C)
–5(t2+ 6)
(t2–3t – 6)2
D)
–5t2
(t2–3t – 6)2
D)
78)
Assume 4
x+y= x. What is the slope of the graph at the point (–1, 9)?
78)
A)
4 +3
B)
3 3
C)
18
D)
30
D)
Differentiate.
79)
f(x) =14x –x3
79)
A)
1
214 –3x2
B)
1
214x –x3
C)
–3x2
14x –x3
D)
14 –3x2
214x –x3
80)
f(x) = (6x + 3)2
80)
A)
36x + 18
B)
72x + 36
C)
12x + 6
D)
36x + 9
81)
f(x) =1
4x – 7
81)
A)
–2
(4x – 7)1/2
B)
–2
(4x – 7)3/2
C)
–1
2(4x – 7)3/2
D)
4
(4x – 7)3/2
Write an equation of the tangent line to the graph of y = f(x) at the point on the graph where x has the indicated value.
82)
f(x) = (5x2+ 5x – 4)(–2x + 1), x = 0
82)
A)
y =13x – 4
B)
y =13x + 4
C)
y =1
13 x + 4
D)
y =1
13 x – 4
16
Differentiate.
83)
y =5x –5
2x2+1
83)
A)
–10x2+ 15x +10
(2x2+1)2
B)
–10x2+20x +5
(2x2+1)2
C)
10x3–20x2+25x
(2x2+1)2
D)
30x2–20x +5
(2x2+1)2
84)
f(x) =(–6x + 5)5
84)
A)
5(–6x + 5)4
B)
–30(–6x + 5)5
C)
–30(–6x + 5)4
D)
–6(–6x + 5)4
85)
f(x) = (3x – 5)( x+ 5)
85)
A)
3
2x+5
2 x
+ 15
B)
9
2x–5
2 x
+ 15
C)
3
2x–5
2 x
+ 15
D)
9
2x+5
2 x
+ 15
86)
Suppose that 15x1/3y–2/3 = 50, where x and y are both differentiable functions of t. Find dy
dt .
86)
A)
25y1/3
2x2/3
B)
10
x2/3y5/3
dx
dt
C)
–y
x
dx
dt
D)
y
2x
dx
dt
Differentiate.
87)
y =6x +9
6x –1
87)
A)
–60
(6x –1)2
B)
48
6x –1
C)
–60x
(6x –1)2
D)
72x +48
(6x –1)2
Solve the problem.
88)
Water is discharged from a pipeline at a velocity v given by v =1660p(1/2), where p is the pressure
(in psi). If the water pressure is changing at a rate of 0.392 psi/second, find the acceleration (dv/dt)
of the water when p =47 psi.
88)
A)
2230.56 ft/sec2
B)
56.9 ft/sec2
C)
47.46 ft/sec2
D)
121.07 ft/sec2
89)
Suppose that x and y are related by the equation x3+(2y + 1)2=y2. Use implicit differentiation to
determine dy
dx .
89)
A)
dy
dx = 3x2+ 2(y + 1)
B)
dy
dx =3x2+ 4(2y + 1)
2y
C)
dy
dx =
–3x2
2(3y + 2)
D)
dy
dx =
–3x2
6y + 1
Solve the problem.
90)
A rectangular steel plate expands as it is heated. Find the rate of change of area with respect to
temperature T when the width is 1.6 cm and the length is 2.5 cm if dl/dt =1.8 x 10–5 cm/°C and
dw/dt =8.2 x 10–6 cm/°C.
90)
A)
2.9 x 10–5 cm2/°C
B)
4 x 10–5 cm2/°C
C)
1.5 x 10–5 cm2/°C
D)
4.9 x 10–5 cm2/°C