Differentiate implicitly to find the slope of the curve at the given point.
91)
x3–y3= 5; (5, 6)
91)
A)
–36
25
B)
–25
36
C)
25
36
D)
25
6
D)
Solve the problem.
92)
Murrel‘s formula for calculating the total amount of rest, in minutes, required after performing a
particular type of work activity for 30 minutes is given by the formula R(w) =30(w – 4)
w – 1.5 , where w
is the work expended in kilocalories per min. A bicyclist expends 5 kcal/min as she cycles home
from work. Find R'(w) for the cyclist; that is, find R'(5).
92)
A)
7.35 min2/kcal
B)
6.12 min2/kcal
C)
8.57 min2/kcal
D)
4.9 min2/kcal
D)
93)
The total revenue for the sale of x items is given by:
R(x) =170 x
7+x3/2 .
Find the marginal revenue R'(x).
93)
A)
R'(x) =85(7x–1/2 – 2x)
7+ x3/2
B)
R'(x) =85(7x–1/2 + 4x)
(7 + x3/2)2
C)
R'(x) =85(7x1/2 – 2x)
(7 + x3/2)2
D)
R'(x) =85(7x–1/2 – 2x)
(7 + x3/2)2
D)
Find dy/dx by implicit differentiation.
94)
2y – x + xy =5
94)
A)
–1 + y
x + 2
B)
1 – y
2 + x
C)
y + 1
x + 2
D)
–1 – y
x + 2
19
Differentiate.
95)
f(x) = (2x3+ 4)(2x7– 5)
95)
A)
8x9+ 56x6– 30x2
B)
8x9+ 56x6– 30x
C)
40x9+ 56x6– 30x2
D)
40x9+ 56x6– 30x
96)
f(x) = (–3x7–x4)(2x2– 5x + 3)
96)
A)
–84x7– 105x6+ 16x4+ 20x3
B)
30x8– 90x7+ 63x6+ 4x5– 15x4+ 12x3
C)
–84x7+ 105x6– 16x4+ 20x3
D)
–54x8+ 120x7– 63x6– 12x5+ 25x4– 12x3
97)
If f(x) =3
x+x3 and g(x) = 1 –x2, find d
dx f(g(x)).
97)
A)
–3
(1 –x2)2+ 3(1 –x2)2
B)
–6
x– 2x3–3
x2+ 3x2
C)
6x
(1 –x2)2– 6x(1 –x2)2
D)
6x
x2– 6x2
Differentiate.
98)
f(x) =(4x2– 7)3
98)
A)
24(4x2– 7)2
B)
3(4x2– 7)2
C)
(24x – 7)(4x2– 7)2
D)
24x(4x2– 7)2
Solve the problem.
99)
A product sells by word of mouth. The company that produces the product has noticed that
revenue from sales is given by R(x) =2 x, where x is the number of units produced and sold. If the
revenue keeps changing at a rate of $800 per month, how fast is the rate of sales changing when
1100 units have been made and sold? (Round to the nearest dollar per month.)
99)
A)
$106,132/month
B)
$13,266/month
C)
$24/month
D)
$26,533/month
Differentiate.
100)
f(x) =x – 2
x + 2
100)
A)
2
(x + 2)2
B)
4
(x – 2)2
C)
2
x + 2
D)
4
(x + 2)2
Differentiate implicitly to find the slope of the curve at the given point.
101)
x3y3=8; (2, 1)
101)
A)
2
B)
4
C)
–1
4
D)
–1
2
102)
y4+x3=y2+11x; (0, 1)
102)
A)
11
2
B)
11
4
C)
11
6
D)
–7
2
21
Differentiate.
103)
h(r) =r2– 5r – 4
5r – 4
103)
A)
5r2– 8r + 40
(5r – 4)2
B)
2r – 5
5
C)
5r2– 6r + 40
(5r – 4)2
D)
5r2– 8r + 40
5r – 4
Find the second derivative of the function.
104)
y =x4+10
x2
104)
A)
2 –60
x4
B)
2x –20
x3
C)
2 +60
x4
D)
1 +60
x4
105)
Assume 4x3+ 2xy –y3=1
2. What is the slope of the graph at the point 1
2, –1 ?
105)
A)
8
5
B)
1
C)
8
D)
–1
2
Solve the problem.
106)
A metal cube dissolves in acid such that an edge of the cube decreases by 0.45 mm/min. How fast is
the volume of the cube changing when the edge is 8.7 mm?
106)
A)
–102 mm3/min
B)
–296 mm3/min
C)
–34 mm3/min
D)
–505 mm3/min
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.
107)
f(x) =4x2+ 2
–4x – 2 , x = 0
107)
A)
y = – 2x + 1
B)
y =2x + 1
C)
y = – 2x – 1
D)
y =2x – 1
Find dy/dx by implicit differentiation.
108)
3xy +2y –8= 0
108)
A)
–3y
3xy +2
B)
–3y
3x +2
C)
–3(x + y)
2
D)
–3y(x + 1)
2
B
Differentiate implicitly to find the slope of the curve at the given point.
109)
x2+ y2= 1; (4, 5)
109)
A)
–4
5
B)
4
5
C)
–5
4
D)
9
5
A
110)
Find the values of x where the tangent line is horizontal for the graph of f(x) =4x2
x + 2 .
110)
A)
x = – 2, x = 0, x = – 4
B)
x = 0, x = – 4
C)
x = – 2
D)
x = 0, x = – 2
B
A function h(x) is defined in terms of a differentiable f(x). Find an expression for h(x).
111)
h(x) = (3x2+5x +9)f(x)
111)
A)
(3x2+5x +9)f(x) + (6x +5)f (x)
B)
(6x +5)f(x) + (3x2+5x +9)f (x)
C)
(3x2+5x +9)f (x)
D)
(6x +5)f (x)
B
23
D
Differentiate.
112)
F(r) =r2(r – 1)(r + 1)–1
112)
A)
(2r3+ 2r2– 2r)(r + 1)–2
B)
–2r
(r + 1)2
C)
2r(r2+ r –1)
D)
4r3– 2r
Provide an appropriate response.
113)
If f(x) and g(x) are differentiable functions, find g(x) given that f(x) =1
x2 and
d
dx f(g(x)) =16x
(8x2+ 5)2.
113)
A)
g(x) =(8x2+ 5)2
B)
g(x) =8x2+ 5
C)
g(x) =16x
8x2+ 5
D)
g(x) =1
8x2+ 5
Express the given function H as a composition of two functions f(x) and g(x) such that H(x) = f(g(x)).
114)
H(x) =6
8x + 9
114)
A)
f(x) =6
x, g(x) =8x + 9
B)
f(x) =8x + 9, g(x) =6
C)
f(x) =6
x, g(x) =8x + 9
D)
f(x) =6, g(x) =8x + 9
Differentiate.
115)
f(x) =(5x + 3)2
115)
A)
10(5x + 3)2
B)
5(5x + 3)
C)
10(5x + 3)
D)
2(5x + 3)
Express the given function H as a composition of two functions f(x) and g(x) such that H(x) = f(g(x)).
116)
H(x) = 3
x2+ 3
116)
A)
f(x) =1
x, g(x) =3
x+ 3
B)
f(x) = x + 3, g(x) =3
x2
C)
f(x) = x, g(x) =3
x+ 3
D)
f(x) =3
x2, g(x) =3
Find dy/dx by implicit differentiation.
117)
y2–x2=2
117)
A)
–y
x
B)
y
x
C)
x
y
D)
–x
y
Solve the problem.
118)
The total profit from selling x units of cookbooks is P(x) = (4x – 4)(6x – 4). Find the marginal
average profit function.
118)
A)
24 –16
x2
B)
24x – 16
C)
24 –40
x2
D)
24x – 40
119)
Let g(x) =x . Using the chain rule, find an expression for the derivative of [g(f(x))].
119)
A)
1
2f(x) f'(x)
B)
x f( x)
C)
x f'(x) +1
2x–1/2 f(x)
D)
1
2 x f'( x)
E)
none of these
Solve the problem.
120)
The formula E = 1000(100 – T) + 580(100 – T)2 is used to approximate the elevation (in meters)
above sea level at which water boils at a temperature of T (in degrees Celsius). Find the rate of
change of E with respect to T for a temperature of 58°C.
120)
A)
–49,720 m/°C
B)
–48,720 m/°C
C)
49,720 m/°C
D)
–83,360 m/°C
Find the maximum.
121)
The graph below shows y =8x
16 +x2 for x 0. Find the coordinates of the maximum point.
121)
A)
16, 8
B)
4, 1
C)
16, 2
D)
4, 2
122)
f(x) =1
4x2+ 6
122)
A)
–8x + 6
(4x2+ 6)2
B)
–8x
4x2+ 6
C)
–8x
(4x2+ 6)2
D)
–1
(4x2+ 6)2
123)
f(x) =2x +5
x –5
4
123)
A)
–60(2x +5)3
(x –5)5
B)
15(2x +5)3
(x –5)5
C)
2x +5
x –5
3
D)
–60
(x –5)2
3
124)
If y = u + 1 –81/3 and u =t
6+ 1, find dy
dt .
124)
A)
–2 ·8–2/3
B)
–t2
6
C)
–1
6t2·8–2/3
D)
1
6
125)
Find the equation of the tangent line to the graph of the function f(x) =x – 3
3x – 5 at x = 1.
125)
A)
y =3
2x –1
2
B)
y = x
C)
y = 3x – 2
D)
y = – 3x + 4
Solve the problem.
126)
The population P, in thousands, of a small city is given by:
P(t) =100t
2t2+3.
where t = the time, in months. Find the growth rate.
126)
A)
P'(t) =100(3 + 6t2)
(2t2+3)2
B)
P'(t) =100(3 – 2t2)
2t2+3
C)
P'(t) =100(2t2–3)
(2t2+3)2
D)
P'(t) =100(3 – 2t2)
(2t2+3)2
Differentiate.
127)
f(x) =2x(3x +2)2
127)
A)
2(3x +2)2(5x +2)
B)
2(3x +2)1
C)
2(9x +2)1
D)
2(3x +2)1(9x +2)
Find dy/dx by implicit differentiation.
128)
1
3x3–4y2=1
128)
A)
8y
x2
B)
x2
4y
C)
8y
x2+1
D)
x2
8y
Differentiate.
129)
y =x2+ 4x(x3/2)
129)
A)
6x(x1/2) + 4x3/2
B)
x3/2 (2x + 4) – (x2+ 4x) 3
2x1/2
(x3/2)x2
C)
2x + 10x3/2
D)
(x2+ 4x) 3
2x1/2 + (x3/2)(2x + 4)
Solve the problem.
130)
A piece of land is shaped like a right triangle. Two people start at the right angle at the same time,
and walk at the same speed along different legs of the triangle while spraying the land. If the area
covered is changing at 5 m2/sec, how fast are the people moving when they are 3 m from the right
angle? (Round
approximations to two decimal places.)
130)
A)
0.83 m/sec
B)
1.80 m/sec
C)
3.33 m/sec
D)
1.67 m/sec
131)
If $5000 is invested at interest rate i, compounded quarterly, it will grow in 3 years to an amount A,
in dollars, given by A =5000 1 +i
4
12. Find the rate of change, dA
di .
131)
A)
dA
di =60,000 1 +i
4
12
B)
dA
di =15,000 1 +i
4
12
C)
dA
di =60,000 1 +i
4
11
D)
dA
di =15,000 1 +i
4
11
Find dy/dx by implicit differentiation.
132)
12x3–x2y3= 7
132)
A)
36x2– 2xy3
3xy2
B)
36x2– 2xy2
3x2y2
C)
36x2– 2xy3
3x2y2
D)
36x2– 2xy2
xy2
Compute f(g(x)) for the given f(x) and g(x).
133)
f(x) =4x6 and g(x) =7
x
133)
A)
4x6
7
B)
4x6
117,649
C)
7
4x6
D)
470,596
x6
Express the given function H as a composition of two functions f(x) and g(x) such that H(x) = f(g(x)).
134)
H(x) = (–4x + 5)5
134)
A)
f(x) = – 4x + 5, g(x) = x5
B)
f(x) = (–4x)5, g(x) =5
C)
f(x) = – 4x5, g(x) = x + 5
D)
f(x) = x5, g(x) = – 4x + 5
Solve the problem.
135)
The concentration of a certain drug in the bloodstream t minutes after swallowing a pill containing
the drug can be approximated using the equation C(t) =1
3(2t + 1) –1/2, where C(t) is the
concentration in arbitrary units and t is in minutes. Find the rate of change of concentration with
respect to time at t =12 minutes.
135)
A)
–1
15 units/min
B)
–1
750 units/min
C)
–1
375 units/min
D)
–1
15 units/min
Differentiate.
136)
y =x
4x –5
136)
A)
–5
(4x –5)2
B)
–5x
(4x –5)2
C)
8x –5
(4x –5)2
D)
–5
4x –5
A function h(x) is defined in terms of a differentiable f(x). Find an expression for h(x).
137)
h(x) =2x2+2x – 1
f(x)
137)
A)
f(x)(4x +2) –f(x)(2x2+2x – 1)
[f(x)]2
B)
f(x)(4x +2) – f(x)(2x2+2x – 1)
[f(x)]2
C)
f(x)(4x +2) –f(x)(2x2+2x – 1)
(2x2+2x – 1)2
D)
f(x)(4x +2) – f(x)(2x2+2x – 1)
f(x)
138)
If f(x) =1
x3+x2, find f(x).
138)
A)
1
(x3+x2)2
B)
–3x + 2
x3(x + 1)2
C)
1
x3+x2
D)
–2x + 1
x2(x + 1)2
Provide an appropriate response.
139)
If f(x) and g(x) are differentiable functions such that f(3) = – 4, f (3) =5, f (4) = – 3, g(3) =4, g (3) =2,
and g (4) = 0, find d
dx f(g(x)) x =3.
139)
A)
–40
B)
10
C)
20
D)
–6
140)
A publishing company has published a new magazine for young adults. The monthly sales S (in
thousands) is given by S(t) =800t
t + 2 , where t is the number of months since the first issue was
published. Find S(3) and S'(3) and interpret the results.
140)
A)
At three months, the monthly sales are $480,000 and decreasing at 64,000 magazines per
month.
B)
At three months, the monthly sales are $480,000 and increasing at 64,000 magazines per
month.
C)
At three months, the monthly sales are $2,400,000 and increasing at 800,000 magazines per
month.
D)
At three months, the monthly sales are $2, 400,000 and increasing at 64,000 magazines per
month.
Find dy/dx by implicit differentiation.
141)
y3+ 3xy + 4x3– 3x = 0
141)
A)
3+ 3y – 12x2
3y2– 3x
B)
3– 3y – 12x2
3y2+ 3x
C)
3– 3y – 12x2
3y2– 3x
D)
3+ 3y – 12x2
3y2+ 3x
32
Answer Key
Testname: C3
33
Answer Key
Testname: C3
Answer Key
Testname: C3
Answer Key
Testname: C3