Find the domain and intercepts.
96)
f(x) =x +36
96)
A)
Domain: [–36, ); y intercept: –36; x intercept: 6
B)
Domain: (–36, ); y intercept: 6; x intercept: –36
C)
Domain: ( , ); y intercept: 6; x intercept: –36
D)
Domain: [–36, ); y intercept: 6; x intercept: –36
Solve the problem.
97)
A 60 room hotel is filled to capacity every night at a rate of $40 per room. The management wants
to determine if a rate increase would increase their profit. They are not interested in a rate decrease.
Suppose management determines that for each $2 increase in the nightly rate, five fewer rooms will
be rented. If each rented room costs $8 a day to service, how much should the management charge
per room to maximize profit?
97)
A)
$45
B)
$50
C)
$42
D)
The management should leave the rate as it is.
Provide an appropriate response.
98)
Find horizontal asymptotes, if any, for f(x) =2x2– 2
4x3– 3 .
98)
A)
y = 0
B)
y =2
3
C)
y =1
2
D)
y = – 2
3
Find the limit, if it exists.
99)
Find lim
x –2
x2+ x – 2
x2– 4 .
99)
A)
3
4
B)
C)
2
D)
0
Find the intervals where f”(x) < 0 ir f”(x) > 0 as indicated.
100)
f”(x) < 0
100)
A)
(–, –2)
B)
(–5, 2)
C)
(–2, )
D)
(–5, 5)
Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the
function is concave up or concave down, and all inflection points.
101)
f(x) =x2
x2+16 , –< x <
101)
37
A)
f is increasing on [0, ) and decreasing
on (–, 0]. f is concave up on (–, ).
f has no inflection points.
B)
f is increasing on (–, 0] and decreasing
on [0, ). f is concave up on –, –4
3
and 4
3, and concave down on
–4
3, 4
3. f has inflection points
at x = – 4
3 and x =4
3.
C)
f is increasing on [0, ) and decreasing
on (–, 0]. f is concave up on –4
3, 4
3
and concave down on –, –4
3 and
4
3, . f has inflection points at
x = – 4
3 and x =4
3.
D)
f is increasing on (–, 0] and decreasing
on [0, ). f is concave up on –, –4
3
and 4
3, and concave down on
–4
3, 4
3. f has inflection points
at x = – 4
3 and x =4
3.
Solve the problem.
102)
A company manufactures and sells x pocket calculators per week. If the weekly cost and demand
equations are given by:
C(x) = 8,000 + 5x
p = 14 –x
4,000 ,0
x
25,000
Find the production level that maximizes profit.
102)
A)
2000 pocket calculators per week
B)
14,000 pocket calculators per week
C)
18,000 pocket calculators per week
D)
8000 pocket calculators per week
Sketch a graph of the function.
103)
f(x) =4x2+ 24x
103)
A)
B)
39
C)
D)
Provide an appropriate response.
104)
Use the first derivative test to determine the local extrema, if any, for the function:
f(x) =3x4–6x2+ 7.
104)
A)
local min at x = 0 and local max at x = – 1 and x = 1
B)
local max at x = – 1 and local min at x = 0 and x = 1
C)
local max at x = 0 and local min at x = – 1 and x = 1
D)
local max at x = 1 and local min at x = 0
Graph the function and locate intervals on which the function is increasing or decreasing, open intervals on which the
function is concave up or concave down, and all inflection points.
105)
f(x) =x4 ex, –< x <
105)
40
A)
f is increasing on (–, –4] and [0, )
and decreasing on [–4, 0]. f is concave
up on (– 2, ) and concave down on
(–, –2). f has an inflection point at x = – 2.
B)
f is increasing on [0, 4] and decreasing
on (–, 0] and [4, ). f is concave up on
(–, 2) and (6, ) and concave down on
(2, 6). f has inflection points at x = 2
and x = 6.
C)
f is increasing on [–4, 0] and decreasing
on (–, –4] and [0, ). f is concave up on
(–6, –2) and concave down on (–, –6) and
(–2, ). f has inflection points at x = – 6
and x = – 2.
41
D)
f is increasing on (–, –4] and [0, )
and decreasing on [–4, 0]. f is concave
up on (–, –6) and (–2, ) and concave
down on (–6, –2). f has inflection points
at x = – 6 and x = – 2.
Provide an appropriate response.
106)
Find the absolute minimum value of f(x) = 5 + 4x +16
x for x > 0.
106)
A)
min f(x) = f(2) = 21
B)
min f(x) = f(1) = 25
C)
min f(x) = f(2) = 13
D)
min f(x) = f(0) = 5
107)
Find the absolute maximum value of f(x) =x4
ex for x > 0. Round your answer to three decimal
places.
107)
A)
4 at x = 4.689
B)
0.7439 at x = 4
C)
4.689 at x = 0
D)
4.689 at x = 4
108)
Concave downward
108)
A)
(–2, ); x = 2
B)
(–5, 2); x = 0
C)
(–5, 5); no inflection points
D)
(–, –2); x = – 2
D
Provide an appropriate response.
109)
Find the critical values and determine the intervals where f(x) is increasing and f(x) is decreasing if
f(x) = 1 +3
x+2
x2.
109)
A)
decreasing on –4
3, 0 ; increasing on , –4
3 and (0, )
B)
increasing on (–4, 0); decreasing on ( , –4) and (0, )
C)
decreasing on (–4, 0); increasing on ( , –4) and (0, )
D)
increasing on –4
3, 0 ; decreasing on , –4
3 and (0, )
D
Solve the problem.
110)
A local office supply store has an annual demand for 20,000 cases of photocopier paper per year. It
costs $3 per year to store a case of photocopier paper, and it costs $80 to place an order. Find the
optimum number of cases of photocopier paper per order.
110)
A)
327 cases
B)
1,066,667 cases
C)
730 cases
D)
1033 cases
D
Decide if the given value of x is a critical number for f, and if so, decide whether the point for x on f is a local minimum,
local maximum, or neither.
111)
f(x) =(x +6)4; x = – 6
111)
A)
Critical number; minimum at (–6 , 0)
B)
Not a critical number.
C)
Critical number but not an extreme point.
D)
Critical number; maximum at (–6 , 0)
Provide an appropriate response.
112)
Find two numbers whose difference is 18 and whose product is a minimum.
112)
A)
1 and 19
B)
36 and 18
C)
0 and 18
D)
9 and –9
D
Find the intervals where the function has the indicated concavity. Give the x coordinates of inflection points.
113)
Concave downward
113)
A)
(–1, 0) , (1, ); x = 0
B)
(–1, 0); no inflection points
C)
(–1, 0) , (1, ); x = 0 and x = 1
D)
(–, –1); no inflection points
A
Find the limit, if it exists.
114)
Find lim
x +
ln x
x .
114)
A)
0
B)
–
C)
1
D)
A
A
Provide an appropriate response.
115)
Find the critical values and determine the intervals where f(x) is decreasing for f(x) =3(x – 4)2/3 + 6.
115)
A)
f(x) is increasing on ( , 4); decreasing on (4, )
B)
f(x) is decreasing on ( , –4); increasing on (–4, )
C)
f(x) is decreasing on ( , 6); increasing on (6, )
D)
f(x) is decreasing on ( , 4); increasing on (4, )
Solve the problem.
116)
Because of material shortages, it is increasingly expensive to produce 6.2L diesel engines. In fact,
the profit in millions of dollars from producing x hundred thousand engines is approximated by
P(x) = – x3+ 30x2+ 10x – 52, where 0
x
20. Find the inflection point of this function to determine
the point of diminishing returns.
116)
A)
(10, 2048)
B)
(10, 1958)
C)
(10, 114.67)
D)
(7.50, 2048)
Answer Key
Testname: C12
Answer Key
Testname: C12
47
Answer Key
Testname: C12
Answer Key
Testname: C12