Download Math120R: Precalculus Test 2 Review, Spring 2017 Sections 2.7

Math120R: Precalculus Test 2 Review, Spring 2017
Sections 2.7, 3.1-3.3, 3.7, 4.1-4.2, Function Modeling
Note: This study aid is intended to help you review for test 2. Do not expect this
review to be identical to the actual test 2 given in your class.
Multiple Choice Practice
1.
2.
Find Q 1 (t ) if Q(t ) 
C
. ( C is a nonzero real number)
4t  1
4t  1
C
(A) Q 1 (t ) 
C
t C
4
(B) Q 1 (t ) 
(D) Q 1 (t ) 
C  4t
,t0
t
(E) None of these
(C) Q 1 (t ) 
C t
,t0
4t
Suppose g (4)  30 means the volume of water in a container is 30 ounces when the depth
of the water is 4 inches. What is the meaning of g 1 (50)  10 ?
(A) The volume of the water is 10 ounces when the depth of the water is 50 inches.
(B) The depth of the water is 10 inches when the volume of the water is 50 ounces.
(C) The depth of the water is 0.2 inches when the volume of the water is 50 ounces.
(D) The volume of the water is 5 ounces when the depth of the water is 10 inches.
(E) None of these
3.
If f (x) is a one-to-one function, and f (8)  11, then which of the following CANNOT
be true?
(A)
f (11)  8
(D)
f
1
(11)  5
(B) f
1
(11)  8
(C) f
1
(5)  3
(E) f (8)  11


4. Using the tables of the functions f and g below, find f  g 1 5
x
0
2
4
6
8
5.
f(x)
5
8
11
14
17
x
0
4
5
6
7
(A)
17
(B) 14
(D)
8
(E) 2
g(x)
2
5
8
11
14
(C) 11
The sum of the base and the height of a triangle is 20 centimeters. Find the maximum area
of the triangle.
(A) 20 square centimeters
(B) 50 square centimeters
(C) 100 square centimeters
(D) 200 square centimeters
(E) 400 square centimeters
6.
Given the graph of the function y  h(t ) , which ONE of the following is the graph of
y  h 1 (t ) ?
y  h(t )
(A)
(C)
(B)
(D)
7.
A rectangle with length L and width W has a diagonal of 10 inches. Express the
perimeter P of the rectangle as a function of L .
(A) P  10L  2L2
(B) P  2L  2 100  L2
(C) P  L( 100  L2 )
(D) P  2L  100
(E) P  L2  W 2  100 .
8. Determine the exact zeros of the function f x   x 2  3x  5 . The exact zeros are:
(A) x 
 3  29
 3  29
or x 
2
2
(B) x  5 or x  2
(C) x  2 or x   2
(D) x  1 or x  4
(E) x 
9.
 3  11
 3  11
or x 
2
2
Find the vertex of the quadratic function f ( x) 
4 2 16
x  x  3 . The y - coordinate
7
7
of the vertex is:
(A)
1
2
(B)
5
7
(C)
6
7
(D) 1
(E) None of these
10. A horticulturist has determined that the number of inches a young oak tree grows in one year
is a function of the annual rainfall r given by g (r )  0.01r 2  0.1r  2. What is the
maximum number of inches a young oak can grow in one year? The maximum number of
inches is:
(A) less than 1
(B) between 1 and 2
(D) between 3 and 4
(E) between 4 and 5
(C) between 2 and 3
11. The graph of y  f (x) is given below. Which ONE of the following could be the correct
equation for f (x) ?
(A)
f x  x  3  9
(B)
f x   
(C)
f x  
(D)
f x  
(E)
f x   
2
1
 x  3 2  9
4
1
x  32  9
4
1
x  12  5
4
1
x  12  5
4
12. Let f ( x)  a( x  b) 3 ( x  c) 4 , where b and c are real numbers and a  0. Which ONE of
the following represents the correct end behavior of f (x) ?
as x  
as x  
(A)
y
y
(B)
y   as x  
y   as x  
(C) y  
as x  
y   as x  
(D)
y  
y
as x  
as x  
13. The graph of y  h(x ) is given below. Which ONE of the following is the correct equation
for h(x) ?
(A) h( x)  3x  1( x  4)
(B) h( x)  
1
x  12 ( x  4)
3
(C) h( x)  3x  12 ( x  4)
(D) h( x)  
1
x  1( x  4)
3
(E) h( x)  3x  12 ( x  4)
14. Suppose the graph of a polynomial function y  f (x) has the following end behavior:
as x  
as x  
y  
y
Which ONE of the following statements must be true?
(A)
The degree of f (x) is an even number.
(B)
The graph of f (x) has no x  intercepts.
(C) The leading coefficient of f (x) is a positive number.
(D)
The domain of f (x) is all positive real numbers.
(E)
The range of f (x) is all real numbers.
15. Find all the real zeros of f ( x)  x 3  5 x 2  7 x  2 . The largest real zero is:
(A)
3 5
2
(B)  0.5
(D)
3 7
2
(E)  2
(C)
 3  13
2
16. What is the remainder when p ( x)  x 4  x 3  x 2  2 is divided by x  3 ?
(A)  26
(B)  17
(C) 0
(D) 43
(E) None of these
17. Let P y   y 2 (5 y  3)( y  6). Solve the inequality P y   0 .
 3 
(A)   ,6 
 5 
 3 
(B)   ,6
 5 
3

(D)   ,   6,  
5

(E) all real numbers
18. Let f r  
(A)
(B)
3

(C)   ,   6,  
5

r2  5
. Which ONE of the following statements is true?
12r 2  7
f (r ) has exactly two vertical asymptotes.
y  0 is a horizontal asymptote.
(C) The domain of f (r ) is all real numbers.
19.
(D)
f (r ) has a slant asymptote.
(E)
f (r ) has exactly one vertical asymptote.
Determine the value of A so that the horizontal asymptote of f x  
is y  10 .
(A)
A  40
(D) A  
1
40
(B) A  
(E)
A0
5
2
(C)
A
2
5
4x 2  5
Ax 2  2
20.
21.
Solve for x :
x 2  3x
0
x 1
(A)
x  0 , x  3 , or x  1 only.
(B)
x  1 only
(C)
x  3 or x  1 only.
(D)
x  3 or x  0 only.
(E)
x  0 only
Find the vertical asymptote(s), if any, for f ( x) 
(A) x  2
(D) x 
22.
5
3
(B) x  0
x2
.
3x 2  5 x
(C) x  0 and x 
5
3
(E) There are no vertical asymptotes.
Find the slant asymptote of f ( x) 
2x 2  7
.
x3
(A) y  2 x  6
(B)
y  2x  1
(D) y  2 x  10
(E) None of these
(C) y  2
23. Let f ( x)  C  b x . Determine constants C and b so that f (1)  10 and f ( 2) 
(A) C  5, b 
1
2
(B) C  10, b 
1
4
(D) C  1, b 
5
4
(E) None of these
(C) C  2, b  5
24. Which ONE of the following is true about the graph of y  5000e 0.0002x  9000 ?
(A) y   as x   and y  9000 as x   .
(B) y   as x   and y  9000 as x   .
(C) y  5000 as x   and y  9000 as x   .
(D) y  0 as x   and y   as x   .
(E) y  5000 as x   and y  0 as x   .
5
.
4
Short Response Practice
1. Let f t  
7
3
and g t   2 .
2t  1
t

Find and simplify g f
1
t  .
F  32
 273.15 is used to determine the corresponding Kelvin measure
1. 8
K of a temperature with Fahrenheit measure F . Find and simplify the inverse of K . Give
a practical interpretation of the inverse of K .
2. The formula K 
3. Let f ( x) 
4.
 3x  7
. Determine the vertical intercept of f
2x  9
1
.
A piece of wire 20 cm long is cut into two pieces. The first is bent into a circle, the second
into a square. Express the combined total area of the circle and square as a function of x ,
where x represents the length of the wire that is bent into a circle.
5.
1
The volume of a right circular cone is given by the formula V  r 2 h. If the volume of a
3
right circular cone is 50 cubic centimeters, express the radius as a function of height.
6.
An airplane manufacturer can produce up to 15 planes per month. The profit made from
the sale of these planes can be modeled by P( x)  0.2 x 2  4 x  3 where P ( x ) is the
profit in hundred thousand of dollars per month and x is the number of planes made
and sold. Based on this model, how many planes should be made and sold to maximize
the profit? What is the maximum profit?
7.
Determine the shortest distance from the ordered pair (1,4) to the graph of y  x 2 .
Approximate your answer to the nearest hundredth.
8.
A rectangle is inscribed in a semicircle with diameter 10 centimeters as shown.
Express the area of the rectangle, A , as a function of the height of the rectangle, h .
Include units in your answer.
h
10 cm
9.
A closed cylindrical can has a volume of 22 cubic inches.
r
h
Express the surface area, A , of the can as a function of its radius r . Your final answer must
be simplified.
10.
An open top rectangular box with a square bottom has a volume of 150 cubic meters. Its
bottom and sides are made from two different materials. It costs 8 dollars per square meter
for the bottom material, and 10 dollars per square meter for the sides. Express the total cost
of building the box in terms of y , where y represents one of the lengths of the bottom
side.
11. A farmer introduces 100 trout into his pond. The population of the trout can be
150t  100
modeled by the function p (t ) 
, where t is time measured in months.
0.04t  1
Find and give an interpretation of the horizontal asymptote of p (t ) .
12. A rational function f (x) has one vertical asymptote x  2 , and one horizontal asymptote,
y  3 . Also, the graph of f x  passes through the ordered pair (5,0) . Determine an
equation for f (x) with these properties.
13. A spherical balloon is being inflated. Suppose the radius of the balloon is increasing at a
rate of 2 centimeters per second.
a. Express the radius r of the balloon as a function of time t .
b. Express the volume V of the balloon as a function of time t .
14. Let f ( x) 
3x 2  5
.
( Bx  1)( x  2)
a. Determine the value of B so that y 
1
is a horizontal asymptote of f (x ).
4
b. Determine the value of B so that f (x) has a slant asymptote. Find the equation of
the slant asymptote.
15. Let f ( x) 
 3x  7
2x  c
Determine the value of c so that x  9 is a vertical asymptote of f (x ).
16. Determine a possible formula for the polynomial function y  g (x) whose graph is shown
below. Leave your answer in factored form.
17.
Let g x   k x 3  k x 2  5 x  1 . Determine the value of k so that x  2 is a factor
of g x .
18. The percent of the United States population that uses the Internet can be modeled by the
73.9
logistic function given by P( y) 
, where y is the number of years after 1995.
1  5.4e 0.415y
a. Using the model above, find the percentage of Internet users in the United States in
2001, rounded to 2 decimal places.
b. Using the model above, find the percentage of Internet users in the United States in
1995, rounded to 2 decimal places.
19. Let f x  
20.
4
e x 1
and g  x  
3x
. Find the domain of
x5
Solve for the indicated variable:
a.
3x  4
 y,
5  7x
for x
b.  3c  172  282  0 , for c
f
 g x .