Solve the problem.
111)
One hundred dollars is deposited in a savings account at 6% interest compounded continuously.
The function defined by f(x) shown in the figure gives the balance in the account after t years. At
what rate (in dollars per year) is the balance growing after 27 years?
111)
A)
$28/year
B)
$8/year
C)
$7/year
D)
$14/year
Find f'(x) at the given value of x.
112)
f(x) =x +5; Find f (11).
112)
A)
1
8
B)
5
8
C)
3
8
D)
5 3
8
41
Solve the problem.
113)
Suppose that the unit price, p, for x units of a product can be illustrated by the given graph. Find
each limit, if it exists:
lim
x
50–
p(x), lim
x
50+
p(x), lim
x
50 p(x), lim
x
75 p(x)
113)
A)
10; 8; 8; 8
B)
10; 8; does not exist; 8
C)
8; 8; does not exist; 8
D)
8; 8; 8; 8
Find all values x = a where the function is discontinuous.
114)
q(x) =x2+9x –7
114)
A)
a = 0
B)
a =9
C)
a =7
D)
Nowhere
42
Estimate the slope of the tangent line to the curve at the given point.
115)
115)
A)
3
B)
3
2
C)
1
2
D)
2
3
Find f'(x) at the given value of x.
116)
f(x) =x3+3; Find f (2).
116)
A)
–12
B)
12
C)
13
D)
15
Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value.
117)
lim
x–2
x2+9x +14
x +2
117)
A)
36
B)
5
C)
Does not exist
D)
9
Decide whether the limit exists. If it exists, find its value.
118)
lim
x
0 f(x)
118)
A)
Does not exist
B)
0
C)
–2
D)
–1
Find the x–values where the function does not have a derivative.
119)
119)
A)
x = 1, x = 2, x = 3
B)
x = 2
C)
x = 1, x = 3
D)
Exists at all points
Decide whether the limit exists. If it exists, find its value.
120)
lim
x–1 f(x)
120)
A)
0
B)
–1
C)
Does not exist
D)
–2
Solve the problem.
121)
Suppose that the total profit in hundreds of dollars from selling x items is given by
P(x) = – x2+10x –21. Find the marginal profit at x =3.
121)
A)
$400 per item
B)
–$600 per item
C)
$0 per item
D)
$600 per item
122)
The blood alcohol level h hours after consumption of 2 ounces of pure ethanol is given by
C(h) =0.55h
h3–h2+ 5 . Find the blood alcohol level as h approaches infinity.
122)
A)
.11
B)
0
C)
D)
.55
Find all values x = a where the function is discontinuous.
123)
f(x) =
5if x <5
x +6if 5
x 10
16 if x >10
123)
A)
Nowhere
B)
a =5
C)
a = – 5
D)
a =10
Suppose the position of an object moving in a straight line is given by the specified function. Find the instantaneous
velocity at time t.
124)
s(t) =t3+5t +9, t =2
124)
A)
26
B)
17
C)
11
D)
9
Solve the problem.
125)
Suppose the demand for a certain item is given by D(p) = – 4p2+ 4p + 6, where p represents the
price of the item. Find D'(p), the rate of change of demand with respect to price.
125)
A)
D'(p) = – 4p + 4
B)
D'(p) = – 4p2+ 4
C)
D'(p) = – 8p2+ 4
D)
D'(p) = – 8p + 4
126)
The size of a population of mice after t months is P = 100(1 + 0.2t + 0.02t2). Find the growth rate at t
=17 months.
126)
A)
44 mice/month
B)
176 mice/month
C)
88 mice/month
D)
188 mice/month
127)
Suppose that the dollar cost of producing x radios is C(x) =800 +40x – 0.2x2. Find the marginal
cost when 35 radios are produced.
127)
A)
$1955
B)
–$1955
C)
$26
D)
$54
46
Find the x–values where the function does not have a derivative.
128)
128)
A)
x = 3
B)
x = 0
C)
x = 0, x = 3
D)
Exists at all points
Decide whether the limit exists. If it exists, find its value.
129)
lim
x
(–1)–
f(x) and lim
x
(–1)+
f(x)
129)
A)
–5, –2
B)
–2, –7
C)
–7, –5
D)
–7, –2
Give an appropriate answer.
130)
Let lim
x
10 f(x) =64. Find lim
x
10
3f(x).
130)
A)
4
B)
10
C)
3
D)
64
Solve the problem.
131)
Suppose that the cost, p, of shipping a 3–pound parcel depends on the distance shipped, x,
according to the function p(x) depicted in the graph. Find each limit, if it exists:
lim
x
100 p(x), lim
x
500 p(x), lim
x
1500 p(x)
131)
A)
5; 5; 15
B)
5; 10; 15
C)
5; does not exist; 15
D)
5; does not exist; does not exist
Give an appropriate response.
132)
Find the limit of f(x) as x approaches 3 from the right.
f(x) =
–2if x <3
x +2if 3 x 5
7if x >5
132)
A)
7
B)
–2
C)
5
D)
The limit does not exist.
C
Complete the table and use the result to find the indicated limit.
133)
If f(x) =x4– 1
x – 1 , find lim
x
1 f(x).
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x)
133)
A)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 1.032 1.182 1.198 1.201 1.218 1.392 ; limit =
B)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 1.032 1.182 1.198 1.201 1.218 1.392 ; limit = 1.210
C)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 3.439 3.940 3.994 4.006 4.060 4.641 ; limit = 4.0
D)
x 0.9 0.99 0.999 1.001 1.01 1.1
f(x) 4.595 5.046 5.095 5.105 5.154 5.677 ; limit = 5.10
Use a graphing utility to find the limit, if it exists.
134)
lim
x
2
x2–4
x2–6x +8
134)
A)
0
B)
Does not exist
C)
– 1
D)
– 2
Use a graphing calculator to find f'(x) when x has the given value.
135)
f(x) = – 7x2+4x; x =17
135)
A)
–248
B)
–242
C)
–115
D)
–234
49
Find the equation of the secant line through the points where x has the given values.
136)
f(x) =4–x2; x = – 4, x =0
136)
A)
y =4x +4
B)
y =4x –4
C)
y =4
D)
y = – 4x –4
Sketch the derivative of the graph.
137)
137)
A)
B)
C)
D)
Find f'(x) at the given value of x.
138)
f(x) =x ; Find f (16).
138)
A)
1
8
B)
16
C)
1
4
D)
4
Find all values x = a where the function is discontinuous.
139)
g(x) =
0if x< 0
x2–2x if 0
x 2
2if x >2
139)
A)
a = 0
B)
a = 0, 2
C)
a =2
D)
Nowhere
Find the average rate of change for the function over the given interval.
140)
y =2x between x = 2 and x = 8
140)
A)
1
3
B)
7
C)
2
D)
–3
10
Find the value of the constant k that makes the function continuous.
141)
h(x) =x2 if x 3
x + k if x >3
141)
A)
k =3
B)
k =6
C)
k = – 3
D)
k =12
Use a graphing calculator to find f'(x) when x has the given value.
142)
f(x) =x6/x; x =3
142)
A)
–0.5917
B)
–0.2357
C)
–0.16441
D)
0.1122