Precalculus
Semester 1: Final Exam Review
Name:
Date:
Friday January 10, 2014
Unit 1, Unit 2, Unit 3… Final Exam Review #2
CALCULATOR OK!
Given f ( x) = x 2 − 5 and g ( x ) =
f ( g ( x))
1.
4.
x + 5 , find the following:
2.
( f − g )(4)
3.
f ( g (20))
A shot-putter throws a ball at an inclination of 45˚to the horizontal. The following data
represents the height of the ball h at the instant that it has traveled x feet horizontally.
a.) Create a scatter plot of the data on your calculator.
b.) Use the scatter plot and your calculator to find a regression equation that best models
the data.
5.
In 2005, hurricane Katrina struck the Gulf Coast of the United States, killing 1289
people and causing an estimated $200 billion in damage. The following data
represent the number of major hurricane strikes in the United States (category 3, 4
or 5) each decade from 1921 to 2000.
a.) Create a scatter plot of the data on your calculator.
b.) Use the scatter plot and your calculator to find a regression equation that best
models the data.
6.
Decade
x
1
2
3
4
5
6
7
8
The following data represent the percentage of the U.S. population aged 25 years or older,
whose age is x, who do not have a high school diploma as of March 2005.
a.) Create a scatter plot of the data on your calculator.
Distance
x
20
40
60
80
100
120
140
160
180
200
220
260
Height
h
25
40
55
65
71
77
77
75
71
64
53
23
Major Hurricanes
In U.S.
5
8
10
8
6
4
5
7
Age
x
30
40
50
60
70
80
% Without
Diploma
16.1
11.6
10.9
13.7
22.3
30.2
b.) Use the scatter plot and your calculator to find a regression equation that best models the data.
7.
The following data represent the temperature T (˚Fahrenheit) in Kansas City, Missouri, x
hours after midnight on May 15, 2005.
a.) Create a scatter plot of the data on your calculator.
b.) Use the scatter plot and your calculator to find a regression equation that best models
the data.
Hours After
Midnight
x
3
6
9
12
15
18
21
24
Temp
(˚F)
45.0
44.1
51.1
57.9
63.0
63.0
59.0
54.0
8.
A sheet of metal 12 inches by 10 inches is to be used to make a open box. Squares of
equal sides x are cut out of each corner then the sides are folded to make the box.
a.) What size should be cut from the metal, x, in order to obtain the maximum
volume?
c.) What size should be cut from the metal, x,
in order to design a box with a volume of
3
at most 60in ?
b.) What size should be cut from the metal, x,
in order to design a box with a volume of
3
at least 60in ?
9.
Suppose in a factory that manufactures cardboard boxes, lidless boxes are formed by cutting out squares from
each corner of a rectangle cardboard with dimensions 8cm x 5cm. In order to utilize the given rectangle most
effectively, the factory needs to know what size squares should be cut off from each corner.
a.) What size should be cut from the cardboard, x, in order to obtain the maximum volume?
b.) What size should be cut from the
cardboard, x, in order to design a box with
3
a volume of at least 10 cm ?
Write the function as a product of linear factors:
10.
f ( x) = x 3 + 5 x 2 + 9 x + 45
12.
f ( x) = 7 x 3 − 17 x 2 − 239 x + 105
c.) What size should be cut from the
cardboard, x, in order to design a box with
3
a volume of at most 10 cm ?
11.
f ( x) = x 3 − x 2 + 25 x − 25
13.
f ( x) = 6 x3 + 28 x 2 − 144 x + 128
Find all of the real zeros of the function:
f ( x) = x 4 + 11x 3 + 11x 2 − 77 x − 126
14.
15.
Solve the inequalities:
16.
x3 + x 2 − 5 x + 3 > 0
17.
x 4 + 3x3 − 3x 2 − 11x − 6 ≥ 0
19.
x3 − 13x 2 + 56 x − 80 ≤ 0
18.
20.
x3 − 8 x 2 + 5 x + 50 ≥ 0
f ( x) = x 4 + 5 x3 − 7 x 2 − 55 x − 44
An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the
object's height s at time t seconds after launch is s ( t ) = –4.9t 2 + 19.6t + 58.8 , where s is in meters. What is
the maximum height above ground level that the object will reach?
21.
You throw a ball to your friend. The ball leaves your hand 5 feet above the ground and has an initial vertical
velocity of 50 feet per second. The throw is modeled by h(t ) = −16t 2 + 50t + 5 . What is the maximum
height above ground level that the ball will reach?
22.
How much would need to been invested in an account that pays 3.5% interest compounded monthly in order to
have a balance of $5000.00 after 5 years?
23.
How much would Jessica need to deposit into an account that pays 2.6% annual interest compounded
quarterly, in order to obtain a balance of $2500 after 4 years?
24.
Fruit flies are placed in a half – pint milk bottle with a banana (for food) and yeast plants (for food and to
provide a stimulus to lay eggs). Suppose that the fruit fly population after t days is given by
P (t ) =
230
1 + 56.5e −0.37t
a.) What is the carrying capacity and the growth rate?
b.) What is the initial population?
c.) What is the population after 5 days?
d.) How long does it take for the population to reach 180?
NO CALCULATOR!
Graph the piecewise functions:
25.
− x 2 + 5, if x < 3
f ( x) =  1
 3 x − 5, if x ≥ 3
2 x + 1,
f ( x) =  2
 x − 3,
26.
6
6
4
4
2
2
-5
5
if x < 1
if x ≥ 1
-5
5
-2
-2
-4
-4
-6
-6
Write the function in vertex form:
27. f ( x) = −3x 2 + 18 x − 24
28. f ( x) = 2 x 2 + 16 x − 83
29. f ( x) = −10 x 2 + 20 x + 38
Solve:
30. log( x + 15) + log x = log 34
31. log ( 5 x ) − log 3 = log 31
32. log ( 3 x ) + log 9 = log15
Expand and simplify completely the logarithmic expression:
33. log 3 27x 2 y
34. log 4 16x 5
 32 x 

 y 
35. log 2 