Course Hero has millions of student submitted documents similar to the one

below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Workbook Precalculus Spring 2007
Compiled by Jerry Morris Sonoma State University
Note to Students: This workbook contains examples and exercises
that will be referred to regularly during class. Please make sure to bring the workbook with you every day that we have class.
Math 107 Workbook
2
Table of Contents
Chapter 1 Functions, Lines, and Change
Section Section Section Section Section 1.1 1.2 1.3 1.4 1.5 Functions and Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Formulas For Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Geometric Properties of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
Chapter 2 Functions, Quadratics, and Concavity
Section Section Section Section Section Section Section 2.1 2.1 2.2 2.4 2.5 2.5 2.6 Introduction Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Domain and Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Concavity Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 3 Exponential Functions
Chapter 3 Algebra Gateway Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Chapter 3 Introduction Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Sections 3.1-3.3 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Sections 3.1-3.3 Exponential Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Section 3.4 Continuous Growth and the Number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 4 Logarithmic Functions
Section Section Section Section 4.1 4.2 4.2 4.3 Logarithms and their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Conversion Between Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Logarithms and Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 The Logarithmic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Chapter 5 Transformations of Functions and Their Graphs
Sections 5.1-5.3 Function Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 Section 5.5 Introduction Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Section 5.5 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Chapter 6 Trigonometric Functions
Section 6.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Section 6.2 Introduction The Sine and Cosine Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48 Section 6.2 The Sine and Cosine Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Section 6.3 Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Section 6.4 Supplement The Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Sections 6.4 & 6.5 Sinusoidal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 Section 6.4 & 6.5 Graphs of Sinusoidal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Section 6.6 Reference Angles Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Section 6.6 Other Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Section 6.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Section 6.7 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Math 107 Workbook
3
Table of Contents
Chapter 7 Trigonometry
Section Section Section Section 7.1 7.2 7.2 7.2 The Laws of Sines and Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Using Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Some Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Using Trigonometric Identities II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter 8 Compositions, Inverses, and Combinations of Functions
Section Section Section Section 8.1 8.1 8.2 8.2 Introduction Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69 Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Introduction Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 9 Polynomial and Rational Functions
Section 9.1 Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Sections 9.2 & 9.3 Introduction Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Sections 9.2 & 9.3 Poynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Section 9.4 & 9.5 Rational Functions I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Section 9.4 & 9.5 Rational Functions II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Appendices
Math 107 Course Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .?? Permission Form for E-mailing of Grade Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
Math 107 Workbook
4
Section 1.1 Functions and Function Notation
Denition. A function is a rule that takes certain values as inputs and assigns to each input value exactly
one output value. 2 . 1+x
Example. Let y = x y 0 1
2
3
Example.
t w = = time (in years) after the year 2000 number of San Francisco 49er victories
t w
0 1 6 12
2 10
3 7
4 2
5 4
6 7
7
Observations:
Math 107 Workbook
5
Example. Which of the graphs below represent y as a function of x?
Graph 1 Graph 2 Graph 3
y
y
y
x
x
x
Example. A woman drives from Aberdeen to Webster, South Dakota, going through Groton on the way,
traveling at a constant speed for the whole trip. (See map below).
20 miles Aberdeen Groton
40 miles Webster
a. Sketch a graph of the womans distance from Webster as a function of time.
b. Sketch a graph of the womans distance from Groton as a function of time.
Math 107 Workbook
6
Section 1.2 Rates of Change
1. Let f (x) = 4 x2 . Find the average rate of change of f (x) on each of the following intervals. (a) 0 x 2 (b) 2 x 4 (c) b x 2b
2. To the right, you are given a graph of the amount, Q, of a radioactive substance remaining after t years. Only the t-axis has been labeled. Use the graph to give a practical interpretation of each of the three quantities that follow. A practical interpretation is an explanation of meaning using everyday language.
Q (grams)
t (yrs) 1 2 3
a. f (1)
b. f (3)
c.
f (3) f (1) 31
Math 107 Workbook 3. Two cars travel for 5 hours along Interstate 5. A South Dakotan in a 1983 Chevy Caprice travels 300 miles, always at a constant speed. A Californian in a 2004 Porsche travels 400 miles, but at varying speeds (see graph to the right).
7
d (miles) 400 300 200 100
1
2 3 t (hours)
4
5
(a) On the axes above, sketch a graph of the distance traveled by the South Dakotan as a function of time. (b) Compute the average velocity of each car over the 5-hour trip.
(c) Does the Californian drive faster than the South Dakotan over the entire 5 hour interval? Justify your answer!
Math 107 Workbook
8
Section 1.3 Linear Functions
1. Let C = 20 0.35t, where C is the cost of a case of apples (in dollars) t days after they were picked. (a) Complete the table below:
t (days) C (dollars)
0
5
10
15
(b) What was the initial cost of the case of apples?
(c) Find the average rate of change of C with respect to t. Explain in practical terms (i.e., in terms of cost and apples) what this average rate of change means.
2. In parts (a) and (b) below, two dierent linear functions are described. Find a formula for each linear function, and write it in slope intercept form. C F 10 50 15 59 20 68 25 77
(a) The line passing through the points (1, 2) and (1, 5).
(b)
Math 107 Workbook
9
3. According to one economic model, the demand for gasoline is a linear function of price. If the price of gasoline is p = $1.10 per gallon, the quantity demanded in a xed period of time is q = 65 gallons. If the price is $1.50 per gallon, the quantity of gasoline demanded is 45 gallons for that period. (a) Find a formula for q (demand) in terms of p (price).
(b) Explain the economic signicance of the slope in the above formula. In other words, give a practical interpretation of the slope.
(c) According to this model, at what price is the gas so expensive that there is no demand?
(d) Explain the economic signicance of the vertical intercept of your formula from part (a).
4. Look back at your answer to problem 2(b). You might recognize this answer as the formula for converting Celsius temperatures to Fahrenheit temperatures. Use your formula to answer the following questions. (a) Find C as a function of F.
(b) What Celsius temperature corresponds to 90 F?
(c) Is there a number at which the two temperature scales agree?
Math 107 Workbook
10
Section 1.4 Formulas For Linear Functions
1. You need to rent a car for one day and to compare the charges of 3 dierent companies. Company I charges $20 per day with an additional charge of $0.20 per mile. Company II charges $30 per day with an additional charge of $0.10 per mile. Company III charges $70 per day with no additional mileage charge. (a) For each company, nd a formula for the cost, C, of driving a car m miles in one day. Then, graph the cost functions for each company for 0 m 500. (Before you graph, try to choose a range of C values would be appropriate.)
(b) How many miles would you have to drive in order for Company II to be cheaper than Company I?
Math 107 Workbook 2. Consider the lines given in the gure to the right. Given that the slope of one of the lines is 2, nd the exact coordinates of the point of intersection of the two lines. (Exact means to leave your answers in fractional form.)
11
3 y
2
x
2
3. Parts (a) and (b) below each describe a linear function. Find a formula for the linear function described in each case. (a) The line parallel to 2x3y = 2 that goes through the point (1, 1). (b) The line perpendicular to 2x 3y = 2 that goes through the point (1, 1).
Math 107 Workbook
12
Section 1.5 Geometric Properties of Linear Functions
Example. Given below are the equations for ve dierent lines. Match each formula with its graph to the right. f (x) = 20 + 2x g(x) = 20 + 4x h(x) = 2x 30 u(x) = 60 x v(x) = 60 2x
E D
x y
A
B
C
Facts about the Line y = mx + b
1. The y-intercept, b (also called the vertical intercept), tells us where the line crosses the 2. If m > 0, the line 3. The larger the value of |m| is, the left to right. If m < 0, the line the graph. left to right. .
Section 2.1 Input and Output Introduction
f(x) 10
a
1. f (10) =
10
.
b
20
x
2. If f (x) = 10, then x = 3. f (a) = 4. f (10) f (6) = .
.
.
Math 107 Workbook
13
Section 2.1 Input and Output
1. The following table shows the amount of garbage produced in the U.S. as reported by the EPA. t (years: 1960 60) G (millions of tons of garbage per year) 60 90 65 105 70 120 75 130 80 150 85 165 90 180
Consider the amount of garbage G as a function of time G = f (t). Include units with your answers. (a) f (60) = (b) f (75) = (c) Solve f (t) = 165.
2. Given is the graph of the function v(t). It represents the velocity of a man riding his bike to the library and going back home after a little while. A negative velocity indicates that he is riding toward his house, away from the library.
20
v (mph)
15 10 5
5 -5 -10 -15
10
15
20
25
30
35
40
45
t (minutes)
Evaluate and interpret: (a) v(5) =
Solve for t and interpret: (d) v(t) = 5
(b) v(40) =
(e) v(t) = 10
(c) v(12) v(7) =
(f) v(t) = v(10)
Math 107 Workbook
14
3. Consider the functions given below. (a) Let f (x) = x2 2x 8. i. Find f (0). (b) Let f (x) = 1 1 x+2
i. Find f (0).
ii. Solve f (x) = 0.
ii. Solve f (x) = 0.
4. Let f (x) =
x . Calculate and simplify f x+1
1 t+1
, writing your nal answer as a single fraction.
Math 107 Workbook
15
Section 2.2 Domain and Range
1. For each of the following functions below, give the domain and the range. f (x)
4
g(x)
4
2
2
4
2 2
2
4
4
2 2
2
4
4
4
2. Oakland Coliseum is capable of seating 63,026 fans. For each game, the amount of money that the Raiders organization makes is a function of the number of people, n, in attendance. If each ticket costs $30.00, nd the domain and range of this function. Sketch its graph.
Math 107 Workbook
16
3. Find the domain and range of each of the following functions. (a) f (x) = 3x + 7 (c) h(x) = x2 x 6
(b) g(x) =
1 (x 1)2
(d) k(x) =
x2 x 6
Math 107 Workbook
17
Section 2.4 Inverse Functions
1. Use the two functions shown below to ll in the blanks to the right. f (x)
4
(a) f (2) = (c) g(0) = (e) f (3) + 1 =
(b) f 1 (2) = (d) g 1 (0) = (f) f 1 (3) + 1 = (h) f 1 (3 + 1) = .
2
4
2 2
2
4
(g) f (3 + 1) =
(g) If g 1 (x) = 0, then x =
4
x g(x) -6 2 -4 0 -2 3 0 7 2 6 4 1 6 5
2. Let A = f (n) be the amount of periwinkle blue paint, in gallons, needed to paint n square feet of a house. Explain in practical terms what each of the following quantities represents. Use a complete sentence in each case. (a) f (20)
(b) f 1 (20)
Math 107 Workbook
18
3. If a cricket chirps R times per minute, then the outside temperature is given by T = f (R) = 1 R + 40 degrees 4 Fahrenheit. (a) Find a formula for the inverse function R = f 1 (T ).
(b) Calculate and interpret f (50) and f 1 (50).
Math 107 Workbook
19
Section 2.5 Concavity Introduction
Denitions. 1. A function f (x) is called increasing if its graph from left to right. graph from left to right. It is called decreasing if its
2. A function f (x) is called concave up if its average rate of change increases from left to right. 3. A function f (x) is called concave down if its average rate of change decreases from left to right. Describe the shape of the graph of a function f (x) that is concave up: Describe the shape of the graph of a function f (x) that is concave down:
Example. Read the following description of a function. Then, decide whether the function is increasing or decreasing. What does the scenario tell you about the concavity of the graph modeling it? When a new product is introduced, the number of people who use the product increases slowly at rst, and then the rate of increase is faster (as more and more people learn about the product). Eventually, the rate of increase slows down again (when most people who are interested in the product are already using it).
Math 107 Workbook
20
Section 2.5 Concavity
1. Consider the functions shown below. Fill in the accompanying tables and then decide whether each function is increasing or decreasing, and whether it is concave up or concave down. (a) Description. This graph gives distance driven as a function of time for a California driver. t d d t 0 2 3 5
d (miles) 400 300 200 100
1
2 3 t (hours)
4
5
(b) Description. This graph gives distance driven as a function of time for a South Dakota driver. t d d t 0 2 3 5
d (miles) 400 300 200 100
1
2 3 t (hours)
4
5
(c) Description. This graph gives the amount of a decaying twinkie as a function of time. t A A t 0 4 6 10
A (ounces) 4 3 2
1
2
4 6 t (years)
8
10
(d) Description. This graph gives the amount of ice remaining in a melting ice cube as a function of time. t A A t 0 4 6 10
A (ounces) 4 3 2
1
2
4 6 t (minutes)
8
10
Math 107 Workbook
21
2. Decide whether each of the following functions are concave up, concave down, or neither. x f (x) 0 1 1 3 2 6 3 10 4 20 x g(x) 0 10 1 9 2 7 3 4 4 0
h(x)
p(x) = 3x + 1
Math 107 Workbook
22
Section 2.6 Quadratic Functions
1. Find (if possible), the zeros of the following quadratic functions. (a) f (x) = x2 + 5x 14 (b) g(x) = x2 + 1
2. The height of a rock thrown into the air is given by h(t) = 40t 16t2 feet, where t is measured in seconds. (a) Calculate h(1) and give a practical interpretation of your answer.
(b) Calculate the zeros of h(t) and explain their meaning in the context of this problem.
(c) Solve the equation h(t) = 10 and explain the meaning of your solutions in the context of this problem.
(d) Use a graph of h(t) to estimate the maximum height reached by the stone. When, approximately, does the stone reach its maximum height? Is the function concave up or concave down?
Math 107 Workbook
23
Chapter 3 Algebra Gateway: Exponents
Evaluate or simplify without a calculator. Write your nal answer in the provided blank. 1. 91/2 + 0.01 =
2.
(xy 3 )2 x0 y 5
=
a3 b1 3. a5/2 b1/2
=
4.
(AB)4 A1 B 2
=
5. 2b1 (b2 + b) 2
=
6.
M + M 1 1 + M 2
=
7. 3 3 t3 + 7(t9 )1/3
=
8.
2km3 + k 2 m km1
=
Solve for x 9. 81x = 3 x=
10.
6 =2 3ax
x=
Math 107 Workbook
24
Chapter 3 Introduction to Exponential Functions
Example 1. The population of a rapidly-growing country starts at 5 million and increases by 10% each year.
Complete the table below:
t (years)
P, population (in millions)
P, increase in population (mil)
0 1 2 3 4
Example 2.
Description The population, P, of ants in your kitchen starts at 10 and increases by 5% per day. The value, V, of a 1982 Chevy Caprice starts at $10000 and decreases by 8% per year. The air pressure, A, starts at millibars at sea level (h = 0) and decreases by % per mile increase in elevation. Growth Factor and Formula
A = 960(0.8)h
Example 3.
Below are the graphs of Q = 150(1.2)t , Q = 50(1.2)t , and Q = 100(1.2)t. Match each formula to the correct graph. Below are the graphs of Q = 50(1.2)t , Q = 50(0.6)t , Q = 50(0.8)t , and Q = 50(1.4)t . Match each formula to the correct graph.
Observations about the graph of Q = abt :
Math 107 Workbook
25
Sections 3.1-3.3 Exponential Functions
1. Suppose we start with 100 grams of a radioactive substance that decays by 20% per year. First, complete the table below. Then, nd a formula for the amount of the substance as a function of t and sketch a graph of the function. t (years) Q (grams) 0 1 2 3 4
2. Suppose you invest $10000 in the year 2000 and that the investment earns 4.5% interest annually. (a) Find a formula for the value of your investment, V, as a function of time.
(b) What will the investment be worth in 2010? in 2020? in 2030?
Math 107 Workbook
26
3. The population of the planet Vulcan and the planet Romulus are recorded in 1980 and in 1990 according to the table below. Also, assume that the population of Vulcan is growing exponentially and that the population of Romulus is growing linearly. Planet Vulcan Romulus 1980 Population (billions) 8 16 1990 Population (billions) 12 20
(a) Find two formulas; one for the population of Vulcan as a function of time and one for the population of Romulus as a function of time. Let t = 0 denote the year 1980.
(b) Use your formulas to predict the population of both planets in the year 2000.
(c) According to your formula, in what year will the population of Vulcan reach 50 billion? Explain how you got your answer.
(d) In what year does the population of Vulcan overtake the population of Romulus? Justify your answer with an accurate graph and an explanation.
Math 107 Workbook
27
Sections 3.1-3.3 Exponential Functions II
1. Find possible formulas for each of the two functions f and g described below. x f (x) 0 2 2 2.5 4 3.125 6 3.90625 g(x)
2
1 3 -1 1
2. Consider the exponential graphs pictured below and the six constants a, b, c, d, p, and q. (a) Which of these constants are denitely positive?
y y=pqx
(b) What of these constants are denitely between 0 and 1?
x y=cd
(c) Which two of these constants are denitely equal?
y=abx x
(d) Which one of the following pairs of constants could be equal? a and p b and d b and q d and q
Math 107 Workbook
28
Section 3.4 Continuous Growth and the Number e
Preliminary Example. At the In-Your-Dreams Bank of America, all investments earn 100% interest
annually. Suppose that you invest $1000 at a time that we will call month 0. Fill in the blank below to compare what your investment will be worth 1 year later using various methods of interest compounding.
Month 0 1 2 3 4 5 6 7 8 9 10 11 12
Compounded 1 Time $1000
Compounded 2 Times $1000
Compounded 4 Times $1000
Alternative Formula for Exponential Functions. Given an exponential function Q = abt , it is possible to rewrite Q as follows: Q= The constant k is then called the continuous growth rate of Q.
Notes:
If k > 0, then Q is increasing. If k < 0, then Q is decreasing.
Math 107 Workbook
29
Exercise Suppose that the population of a town starts at 5000 and grows at a continuous rate of 2% per year.
(a) Write a formula for the population of the town as a function of time, in years, after the starting point.
(b) What will the population of the town be after 10 years?
(c) By what percentage does the population of the town grow each year?
Math 107 Workbook
30
Section 4.1 Logarithms and Their Properties
1. Solve each of the following equations for x. (a) 5 9x = 10 (d) 5x9 = 10
(b) 10e4x+1 = 20
(e) e2x + e2x = 1
(c) a bt = c d2t
(f) ln(x + 5) = 10
Math 107 Workbook
31
2. Simplify each of the following expressions. (a) log(2A) + log(B) log(AB) (b) ln(abt ) ln((ab)t ) ln a
3. Decide whether each of the following statements are true or false. (a) ln(x + y) = ln x + ln y
(b) ln(x + y) = (ln x)(ln y)
(c) ln(ab2 ) = ln a + 2 ln b
(d) ln(abx ) = ln a + x ln b
(e) ln(1/a) = ln a
Math 107 Workbook
32
Section 4.2 Conversion Between Bases
Exercise. Fill in the gaps in the chart below, assuming that t is measured in years:
Formula Q = abt Q = aekt Q = 6e0.04t Q = 5(1.2)t Q = 10(0.91)t
Growth or Decay Rate Per Year Continuous Per Year
Math 107 Workbook
33
Section 4.2 Logarithms and Exponential Models
1. (Taken from Connally) Scientists observing owl and hawk populations collect the following data. Their initial count for the owl population is 245 owls, and the population grows by 3% per year. They initially observe 63 hawks, and this population doubles ever 10 years. (a) Find a formula for the size of the population of owls and hawks as functions of time.
(b) When will the populations be equal?
Math 107 Workbook
34
2. Find the half-lives of each of the following substances. (a) Tritium, which decays at an annual rate of 5.471% per year. (b) Vikinium, which decays at a continuous rate of 10% per week.
3. If 17% of a radioactive substance decays in 5 hours, how long will it take until only 10% of a given sample of the substance remains?
Math 107 Workbook
35
Section 4.3 The Logarithmic Function
1. Consider the functions f (x) = ln x and g(x) = log x. (a) Complete the table below.
x ln x log x
0.1
0.5
1
2
4
6
8
10
(b) Plug a few very small numbers x into ln x and log x (like 0.01, 0.001, etc.) What happens to the output values of each function?
(c) If you plug in x = 0 or negative numbers for x, are ln x and log x dened? Explain.
(d) What is the domain of f (x) = ln x? What is the domain of g(x) = log x?
(e) Sketch a graph of f (x) = ln x below, choosing a reasonable scale the on x and y axes. Does f (x) have any vertical asymptotes? Any horizontal asymptotes?
y
x
Math 107 Workbook
36
2. What is the domain of the following four functions? (a) y = ln(x2 ) (b) y = (ln x)2 (c) y = ln(ln x) (d) y = ln(x 3)
3. Consider the exponential functions f (x) = ex and g(x) = ex . What are the domains of these two functions? Do they have any horizontal asymptotes? any vertical asymptotes?
Math 107 Workbook
Sections 5.1-5.3 Function Transformations
37
1. Consider the function f (x) = x2 4x + 4. Transformation Formula Graph
4 3 2 1 4 3 2 1 1 2 3 1 2 3 4
Description
y = f (x) + 2
4 4 3 2 1 4 3 2 1 1 2 3 1 2 3 4
y = f (x) 2
4 4 3 2 1 4 3 2 1 1 2 3 1 2 3 4
y = f (x + 2)
4 4 3 2 1 4 3 2 1 1 2 3 1 2 3 4
y = f (x 2)
4 4 3 2 1 4 3 2 1 1 2 3 1 2 3 4
y = f (x)
4 4 3 2 1 4 3 2 1 1 2 3 1 2 3 4
y = f (x)
4
Math 107 Workbook 2. Let y = f (x) be the function whose graph is given below. Fill in the entries in the table below, and then sketch a graph of the transformations y = 2f (x) and y = f (x 2).
38
4 2
6
4
2 2 4
2
4
6
x f (x) 2f (x) f (x 2)
-4
-2
0
2
4
6
Math 107 Workbook 3. Let y = f (x) be the function whose graph is given below. Fill in the entries in the table below, and then sketch a graph of the transformations y = f (x) and y = 1 2f (x).
39
4 2
6
4
2 2 4
2
4
6
x f (x) f (x) 1 2f (x)
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Math 107 Workbook 4. Given to the right is the graph of the function x 1 y= . On the same set of axes, sketch the 2 x2 x 1 1 graph of y = and y = 2. 2 2
40
4 3 2 1
4
3
2
1 1 2
1
2
3
4
5. Let H = f (t) be the temperature of a heated oce building t hours after midnight. (See diagram to the right fora graph of f.) Write down a formula for a new function that matches each story below. (a) The manager decides that the temperature should be lowered by 5 degrees throughout the day.
H (degrees F) 80 60 40 20
4
8
12 16 t (hours)
20
24
(b) The manager decides that employees should come to work 2 hours later and leave 2 hours later.
Math 107 Workbook Denition We say that a function is even if f (x) = f (x) for all x in the domain of the function. In other words, an even function is symmetric about the . Sketch
41
We say that a function is odd if f (x) = f (x) for all x in the domain of the function. In other words, an odd function is symmetric about the .
6. Use algebra to show that f (x) = x4 2x2 + 1 is an even function and that g(x) = x3 5x is an odd function.
Math 107 Workbook 7. Given the graph of y = f (x) given below, sketch the graph of the following related functions:
42
f(x)
(a) y = f (2 x) + 2
(b) y = 2 f (1 x)
Math 107 Workbook
43
Section 5.5 Introduction
f (x) = x2
4 3 4 4
3
3
2
2
2
1
1
1
-4
-3
-2
-1 -1
1
2
3
4
-4
-3
-2
-1 -1
1
2
3
4
-4
-3
-2
-1 -1
1
2
3
4
-2
-2
-2
-3
-3
-3
-4
-4
-4
4
4
3
3
2
2
1
1
-4
-3
-2
-1 -1
1
2
3
4
-4
-3
-2
-1 -1
1
2
3
4
-2
-2
-3
-3
-4
-4
Information about Quadratic Functions
In general, a quadratic function f can be written in several dierent ways: 1. f (x) = ax2 + bx + c 2. f (x) = a(x r)(x s) 3. f (x) = a(x h)2 + k Notes. The graph of a quadratic function is called a In factored form, the numbers r and s represent the In vertex form, the point (h, k) is called the . The graph opens upward if . of f. of the parabola. The axis of symmetry is the line and downward if . (standard form, where a, b, and c are constant) (factored form, where a, r, and s are constant) (vertex form, where a, h, and k are constant)
Math 107 Workbook
44
Section 5.5 The Family of Quadratic Functions
1. For each of the following, complete the square in order to nd the vertex. In part (b), your answer will contain the constant b. (a) y = x2 40x + 1 (b) y = 2x2 + bx + 3
2. Find a formula for the quadratic function shown below. Also nd the vertex of the function.
1
-1
2
Math 107 Workbook
45
3. A parabola has its vertex at the point (2, 3) and goes through the point (6, 11). Find a formula for the parabola.
4. (Taken from Connally) A tomato is thrown vertically into the air at time t = 0. Its height, d(t) (in feet), above the ground at time t (in seconds) is given by d(t) = 16t2 + 48t. (a) Find t when d(t) = 0. What is happening to the tomato the rst time that d(t) = 0? The second time?
(b) When does the tomato reach its maximum height? How high is the tomatos maximum height?
Math 107 Workbook
Section 6.1 Periodic Functions
46
Denition. A function f is called periodic if its output values repeat at regular intervals. Graphically,
this means that if the graph of f is shifted horizontally by p units, the new graph is identical to the original. Given a periodic function f : 1. The period is the horizontal distance that it takes for the graph to complete one full cycle. That is, if p is the period, then f (t + p) = f (t). 2. The midline is the horizontal line midway between the functions maximum and minimum output values. 3. The amplitude is the vertical distance between the functions maximum value and the midline.
1. The Brown County Ferris Wheel has diameter 50 meters and completes one full revolution every two minutes. When you are at the lowest point on the wheel, you are still 5 meters above the ground. Assuming you board the ride at t = 0 seconds, sketch a graph of your height, h = f (t), as a function of time.
h (meters) 60
40
20
t (seconds) 60 120 180 240
What are the amplitude, midline and period of the function h = f (t)?
Math 107 Workbook
47
2. The function given below models the height, h, in feet, of the tide above (or below) mean sea level t hours after 6:00 a.m.
20
(a) Is the tide rising or falling at 7:00 a.m.?
10 h
6
(b) When does low tide occur?
10
12 t
24
20
(c) What is the amplitude of the function? Give a practical interpretation of your answer.
(d) What is the midline of the function? Give a practical interpretation of your answer.
3. Which of the following functions are periodic? For those that are, what is the period?
3 2 1 5 4 3 2 1 1 -1 -2 -3 2 3 4 5 6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 1 2 3 4 5
1.5
4
1
0.5
4 4
4
-6
-4
-2
2
4
6
Math 107 Workbook
48
Section 6.2 Introduction
Angle Measurement in Circles
Angles start from the positive x-axis. Counterclockwise dened to be positive.
y
x
Denition. The unit circle is the term used to describe
a circle that has its center at the origin and has radius equal to 1. The cosine and sine functions are then dened as described below.
(1,0)
(0,1)
(1,0)
(0,1)
Theorem. Consider a circle of radius r centered at the origin. Then the x and y coordinates of a point on this circle are given by the following formulas:
(0,r)
(r,0)
(r,0)
(0,r)
Math 107 Workbook
Section 6.2 The Sine and Cosine Function
1. Use the unit circle to the right to estimate each of the following quantities to the nearest 0.05 of a unit. (a) sin(90 ) = (c) sin(180 ) = (e) cos(45 ) = (g) cos(70 ) = (i) sin(100 ) = (b) cos(90 ) = (d) cos(180 ) = (f) sin(90 ) = (h) sin(190 ) = (j) cos(100 ) =
-1 -1 1 1
49
2. For each of the following, ll in the blank with an angle between 0 and 360, dierent from the rst one, that makes the statement true. (a) sin(20 ) = sin( ) (b) sin(70 ) = sin( ) (c) sin(225 ) = sin( )
(d) cos(20 ) = cos(
)
(e) cos(70 ) = cos(
)
(f) cos(225 ) = cos(
)
3. Given to the right is a unit circle. Fill in the blanks with the correct answer in terms of a or b. (a) sin( + 360 ) = (b) sin( + 180 ) = (c) cos(180 ) = (d) sin(180 ) = (e) cos(360 ) = (f) sin(360 ) = (g) sin(90 ) =
y
(a,b) 1
x
4. Use your calculator to nd the coordinates of the point P at the given angle on a circle of radius 4 centered at the origin. (a) 70
(b) 255
Math 107 Workbook
50
Section 6.3 Radian Measure
1. In the pictures below, you are given the radius of a circle and the length of a circular arc cut o by an angle . Find the degree and radian measure of .
8
2
4
4
2. In the pictures below, nd the length of the arc cut o by each angle.
2/3
2
80
3
3. A satellite orbiting the earth in a circular path stays at a constant altitude of 100 kilometers throughout its orbit. Given that the radius of the earth is 6370 kilometers, nd the distance that the satellite travels in completing 70% of one complete orbit.
4. An ant starts at the point (0,3) on a circle of radius 3 (centered at the origin) and walks 2 units counterclockwise along the arc of a circle. Find the x and the y coordinates of where the ant ends up.
Math 107 Workbook
51
Section 6.4 Supplement The Unit Circle
cos
0 0
30
6
45
4
60
3
90
2
120
2 3
135
3 4
150
5 6
180
sin
cos
210
7 6
225
5 4
240
4 3
270
3 2
300
5 3
315
7 4
330
11 6
2 360
sin
1
-3 -2 -1 -1
1
2
3
4
5
6
7
8
9
Math 107 Workbook
52
Sections 6.4 and 6.5 Sinusoidal Functions
Directions. Make sure that your graphing calculator is set in radian mode. Function y = 2 sin x y = sin x + 2 y = sin(x + 2) y = sin(2x) Eect on y = sin x B 1 2 4 1/2 B y y y y y y = sin(Bx) = sin x = sin(2x) = sin(4x) = sin(x/2) = sin(Bx) Period
Summary
For the sinusoidal functions y = A sin(B(x h)) + k and y = A cos(B(x h)) + k: 1. Amplitude = 2. Period = 3. Horizontal Shift = 4. Midline:
Primary Goal in Section 6.5. Find formulas for sinusoidal functions given graphs, tables, or verbal
descriptions of the functions.
Helpful Hints in Finding Formulas for Sinusoidal Functions
1. If selected starting point occurs at the midline of the graph, use the sine function. 2. If selected starting point occurs at the maximum or minimum value of the graph, use the cosine function. 3. Changing the sign of the constant A reects the graph of a sinusoidal function about its midline.
Math 107 Workbook
53
Sections 6.4 and 6.5 Graphs of Sinusoidal Functions
1. Find a possible formula for each of the following sinusoidal functions.
6 2
1
3
-4 -3 -2 -1 -1
1
2
3
4
5
6
7
8
9
-2 2 3 4
-3
3
0.8
2
4
7
-3
-0.8
3
10 8
1 6 -1 2 4
-3
2
11
Math 107 Workbook
54
2. For each of the following, nd the amplitude, the period, the phase shift, and the midline. (a) y = 2 cos(x +
2 3 )
1
(b) y = 3 sin(2x 7)
3. A population of animals oscillates annually from a low of 1300 on January 1st to a high of 2200 on July 1st, and back to a low of 1300 on the following January. Assume that the population is well-approximated by a sine or a cosine function. (a) Find a formula for the population, P, as a function of time, t. Let t represent the number of months after January 1st. (Hint. First, make a rough sketch of the population, and use the sketch to nd the amplitude, period, and midline.)
(b) Estimate the animal population on May 15th.
(c) On what dates will the animal population be halfway between the maximum and the minimum populations?
Math 107 Workbook
55
Section 6.6 Reference Angles Supplement
Denition. The reference angle associated with an angle is the acute angle (having positive measure) formed by the x-axis and the terminal side of the angle .
Key Fact. If is any angle and is the reference angle, then
sin cos tan csc sec cot = = = = = = sin cos csc sec
tan
cot ,
where the correct sign must be chosen based on the quadrant of the angle .
Example. For each of the following angles, sketch the angle and nd the reference angle.
(1) = 300 (2) =
4 3
(3) = 135
(4) =
7 6
Math 107 Workbook
56
Section 6.6 Other Trigonometric Functions
3 1. Suppose that sin = 4 and that 3 2
2. Find the exact values of cos and sec .
2. Suppose that csc =
x 2
and that lies in the 2nd quadrant. Find expressions for cos and tan in terms of x.
Math 107 Workbook 3. Given to the right is a circle of radius 2 feet (not drawn to scale). The length of the circular arc s is 2.6 feet. Find the lengths of the segments labeled u, v, and w. Give all answers rounded to the nearest 0.001.
57
2 v u
s w
Math 107 Workbook
Section 6.7 Inverse Trigonometric Functions
Preliminary Idea.
sin(/6) = 1/2 means the same thing as .
58
Denition.
1. sin1 x is the angle between and 2 2. tan1 x is the angle between and 2
2 2
whose sine is x. whose tangent is x.
3. cos1 x is the angle between 0 and whose cosine is x.
Note. sin1 x, cos1 x, and tan1 x can also be written as arcsin x, arccos x, and arctan x,
respectively.
Exercise. Calculate each of the following exactly.
1. cos
1
3 2
=
2. sin1
2 2
=
3. tan1 ( 3) = 4. sin1 (1) =
Question. How would we nd all solutions to the equation sin x =
1 2
that lie between 0 and 2?
Math 107 Workbook
59
x sin x 0 0 /6 1/2 /4 2/2 /3 3/2 /2 1
x
sin1 x
1.5
1
R
0.5
S
Q P O
0.5 1 1.5
1.5
1
0.5 0.5
1
1.5
x tan x 0 0 /6 3/3 /4 1 /3 3 /2 undened
x
tan1 x
2
R
1
Q P
O
2 1 1 2
1
2
Math 107 Workbook
60
Section 6.7 Solving Trigonometric Equations
1. Solve each of the following trigonometric equations, giving all solutions between 0 and 2. Give exact answers whenever possible. (a) sin =
3 2
1 (b) tan = 4
(c) cos =
1 2
Math 107 Workbook
61
2. Find all solutions to 2 sin x cos x + cos x = 0 that lie between 0 and 2. Give your answers exactly.
3. Use the graph to the right to estimate the solutions to the equation cos x = 0.8 that lie between 0 and 2. Then, use reference angles to nd more accurate estimates of your solutions.
y = cos x
1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 1 2 3 4 5 6 7 8 9
Math 107 Workbook
Section 7.1 The Laws of Sines and Cosines
Right Triangles
62
(0,r)
sin =
cos
=
tan
=
(r,0)
Warmup example. A kite yer wondered how high her kite was ying. She used a protractor to measure an angle of 40 from level ground to the kite string. If she used a full 100-yard spool of string, how high is the kite?
General Triangles: The following formulas hold for any triangle, labeled as shown below.
Law of Sines:
a
C b A
B
Law of Cosines:
c
General Rule. The Law of Cosines can be used when 2 sides of a triangle and the angle in between the sides are known.
Math 107 Workbook
63
Example. Find all possible triangles with a = 3, b = 4, and A = 35 .
Math 107 Workbook
64
Section 7.1 The Laws of Sines and Cosines
1. Two re stations are located 25 miles apart, at points A and B. There is a forest re at point C. If CAB = 54 and CBA = 58 , which re station is closer? How much closer? (Taken from Connally, et. al.)
2. A triangular park is bordered on the south by a 1.7-mile stretch of highway and on the northwest by a 4-mile stretch of railroad track, where 33 is the measure of the acute angle between the highway and the railroad tracks. As a part of a community improvement project, the city wants to fence the third side of the park and seed the park with grass. (a) How much fence is needed for the third side of the park?
(b) What is the degree measure of the angle on the southeast side of the park?
(c) For how much total area will they need grass seed?
Math 107 Workbook 3. To measure the height of the Eiel Tower in Paris, a person stands away from the base and measures the angle of elevation to the top of the tower to be 60 . Moving 210 feet closer, the angle of elevation to the top of the tower is 70 . How tall is the Eiel Tower? (Taken from Connally, et. al.)
65
4. Two points P and T are on opposite sides of a river (see sketch to the right). From P to another point R on the same side is 300 feet. Angles P RT and RP T are found to be 20 and 120 , respectively. (Taken from Cohen.) (a) Compute the distance from P to T.
R
P
T
(b) Assuming that the river is reasonably straight, calculate the shortest distance across the river.
Math 107 Workbook
66
Section 7.2 Using Trigonometric Identities
Directions. On a separate sheet of paper, simplify each of the following as indicated by the instructions. Problem 1 2 3 4 5 6 Starting Expression sec cos 1 + cos sin + 1 + cos sin cos 2t cos t + sin t sin A(csc A sin A) 1 cos2 cos cot 1 cot + 1 Simplication Instructions Simplify so that you nal answer contains two trig functions, no fractions, and no addition or subtraction. Simplify so that your nal answer contains only one trig function and no fractions. Simplify so that your nal answer is a sum of two trig functions with no fractions. Simplify so that your nal answer is a single trig function raised to a power. Simplify so that you nal answer has no fractions and is a product of two trig functions. Simplify so that your nal answer is an expression similar to the one on the left, but contains only tangent functions.
Section 7.2 Some Trigonometric Identities
sin(2) cos(2) cos(2) cos(2) tan(2) = 2 sin cos = 1 2 sin2 = 2 cos2 1 2 tan 1 tan2 = cos2 sin2 = sin() = cos() = tan() = sin cos = = sin cos tan 2 sin + 2 cos
Math 107 Workbook
67
Section 7.2 Using Trigonometric Identities II
1. Given the right triangle to the right, write each of the following quantities in terms of h. Note. Asking you to write something in terms of h is NOT asking you to solve for h. It is asking you to rewrite the quantity that you are given so that h is the only unknown in your answer.
h
1
(a) sin
(b) cos
(c) cos( ) 2
(d) sin(2)
(e) cos(sin1 h)
Math 107 Workbook
68
2. By starting with one side and showing that it is equal to the other side, prove the following trigonometric identity: 1 + cos t sin t = 1 cos t sin t
Math 107 Workbook
69
Section 8.1 Function Composition Introduction
The function h(t) = f (g(t)) is called the composition of f with g. The function h is dened by using the output of the function g as the input of f. Exercise. Complete the table below. t f (t) g(t) f (g(t)) g(f (t)) 0 2 3 3 1 3 0 1 1 2 2 1 0 2 1 3
Section 8.1 Function Composition
1. Given to the right are the graphs of two functions, f and g. Use the graphs to estimate each of the following. (a) g(f (0)) = (b) f (g(0)) =
4
f(x)
(c) f (g(3)) =
(d) g(g(4)) =
4
4
g(x)
(e) f (f (1)) =
4
2. For each of the following functions f (x), nd functions u(x) and v(x) such that f (x) = u(v(x)). (a) 1+x (b) sin(x3 + 1) cos(x3 + 1)
Math 107 Workbook (c) 3
2x+1
1 (d) 1+
70
2 x
3. Let f (x) =
1 . 1 + 2x
(a) Solve f (x + 1) = 4 for x.
(b) Solve f (x) + 1 = 4 for x.
(c) Calculate f (f (x)) and simplify your answer.
Math 107 Workbook
71
4. For each of the following functions, calculate f (x + h) f (x) h and simplify your answers. (a) f (x) = x2 + 2x + 1 (b) f (x) =
1 x
(c) f (x) = 3x + 1
Math 107 Workbook
72
Section 8.2 Inverse Functions Introduction
Denition. Suppose Q = f (t) is a function with the property that each value of Q determines exactly one value of t. Then f has an inverse function, f 1 , and f 1 (Q) = t if and only if Q = f (t).
If a function has an inverse, it is said to be invertible. Example. Given below are values for a function Q = f (t). Fill in the corresponding table for t = f 1 (Q).
t 0 1 2 3 4 f (t) 2 5 7 8 11 Q 2 5 7 8 11 f (Q)
1
Question. Does the function f (x) = x2 have an inverse
function?
f (x) = x2
4 3 2 1 2 1 1 2
Horizontal Line Test. A function f has an inverse function if and only if the graph of f intersects
any horizontal line at most once. In other words, if any horizontal line touches the graph of f in more than one place, then f is not invertible.
Math 107 Workbook
73
Section 8.2 Inverse Functions
1. Find a formula for the inverse function of each of the following functions. (a) f (x) = x1 x+1
(b) g(x) = ln(3 x)
2. Given to the right is the graph of the functions f (x) and g(x). Use the function to estimate each of the following. (a) f (2) = (b) f 1 (2) =
4 g(x)
(c) f 1 (g(1)) =
(d) g 1 (f (3)) =
4 f(x)
4
4
(e) Rank the following quantities in order from smallest to largest: f (1), f (2), f 1 (1), f 1 (2), 0
Math 107 Workbook
74
3. Let f (x) = 10e(x1)/2 and g(x) = 2 ln x 2 ln 10 + 1. Show that g(x) is the inverse function of f (x).
4. Let f (t) represent the amount of a radioactive substance, in grams, that remain after t hours have passed. Explain the dierence between the quantities f (8) and f 1 (8) in the context of this problem.
Math 107 Workbook
75
Section 9.1 Power Functions
Denition. A power function is a function of the form f (x) = kxp , where k and p are constants.
I. Positive Integer Powers. Match the following functions to the appropriate graphs below: y = x2 , y = x3 , y = x4 , y = x5
-1
1
-1
1
II. Negative Integer Powers. Match the following functions to the appropriate graphs below: y = x2 , y = x3 , y = x4 , y = x5
-1 -1 1
1
III. Positive Fractional Powers. Match the following functions to the appropriate graphs below: y = x1/2 , y = x1/3
1
Math 107 Workbook
76
1. (Adapted from Connally) The blood circulation time (t) of a mammal is directly proportional to the 4th root of its mass (m). If a hippo having mass 2520 kilograms takes 123 seconds for its blood to circulate, how long will it take for the blood of a lion with body mass 180 kg to circulate?
2. Find a formula for the power function g(x) described by the table of values below. Be as accurate as you can with your rounding.
x g(x)
2 4.5948
3 7.4744
4 10.5561
5 13.7973
Math 107 Workbook
77
Section 9.2/9.3 Polynomials Introduction
Denition. A polynomial is a function of the form
y = p(x) = an xn + an1 xn1 + + a1 x + a0 , where n is a positive integer and a0 , a1 , . . . , an are all constants. The integer n is called the degree of the polynomial.
Fact 1. A polynomial of degree n can have at most n 1 turnaround points.
Fact 2. As x and x , the highest power of x takes over. (Note. The symbol means approaches.)
Math 107 Workbook will have an even number of (x a) factors.
78
Fact 3. When a polynomial touches but does not cross the x axis at x = a, the factored form of the polynomial Example. Consider the polynomial
p(x) = (x + 3)(x + 2)2 (x + 1)(x 1)(x 2)2 (x 3)2 , whose graph is shown to the right.
3 2 600 400 200 1 200 400 1 2 3 4
number an is called the leading coecient of p, and the number a0 is called the constant coecient of p.
Denition. Let p(x) = an xn + an1 xn1 + + a1 x + a0 be a polynomial such that an = 0. Then the
Example. Find the leading coecient and the constant coecient of each of the following.
1. p(x) = 3x4 5x2 + 6x 1 2. q(x) = x2 (x 3) 3. r(x) = (2x 3)2 (x + 4)
Math 107 Workbook
79
Sections 9.2 and 9.3 Polynomials
Each of the following gives the graph of a polynomial. Find a possible formula for each polynomial. In some cases, more than one answer is possible. 1. 2.
-1
3
-3
-2
2
-3 -24
3.
4.
-3
-2
2
-6
-4
4
5.
6.
-4
1
3
-1
1
3
Also given: (-2,-4) is on the graph
Also given: (2,1) is on the graph
Math 107 Workbook
80
For problems 7 through 10, use your calculator to graph the polynomial on a good viewing window. Then, answer each of the following questions. a. How many roots (zeros) does the polynomial have? b. How many turning points does the polynomial have? 7. y = 2x + 3 8. y = x2 x 2 9. y = x3 2x2 x + 2 10. y = 5x2 + 4 For problems 11 through 15, answer the following questions about the given polynomial: a. What is its degree? b. What is its leading coecient? c. What is its constant coecient? d. What are the roots of the polynomial? First, give your answer(s) in exact form; then, give decimal approximations if appropriate. 11. p(x) = x2 3x 28 12. p(x) = 8 7x 13. p(x) = x(2 + 4x x2 ) 14. p(x) = 2x2 + 4 15. p(x) = (x 3)(x + 5)(x 37)(2x + 4)x2 For problems 16 through 18, answer the following questions about the given polynomial: a. What happens to the output values for extremely positive values of x? b. What happens to the output values for extremely negative values of x? 16. p(x) = 2x3 + 6x 2 17. p(x) = 2x x2 18. p(x) = x6 x 2
19. For each of the following, give a formula for a polynomial with the indicated properties. a. A sixth degree polynomial with 6 roots. b. A sixth degree polynomial with no roots.
Math 107 Workbook
Sections 9.4 and 9.5 Rational Functions I
Denition. A rational function is a function r(x) of the form r(x) =
p(x) , where p(x) and q(x) are q(x) polynomials. In other words, a rational function is a polynomial divided by a polynomial.
81
3x2 + 2x 1 . First, ll in the 2x2 + 1 table to the right for the function f (x). Then, sketch a graph of f (x) in the space below.
Example. Let f (x) =
x f (x)
1
10
100
1000
10000
Example. Algebraically check each of the following for horizontal asymptotes.
(a) f (x) = 3x2 + 2x 1 2x2 + 1 (b) g(x) = 2x + 4 2x2 + 1 (c) h(x) = x6 + 5x3 2x2 + 1 x4 + 2
Math 107 Workbook
82
p(x) q(x)
Finding horizontal asymptotes. Let f (x) =
be a rational function.
1. If the degree of p(x) equals the degree of q(x), then f (x) has a horizontal asymptote at y= leading coecient of p(x) . leading coecient of q(x)
2. If the degree of p(x) is less than the degree of q(x), then y = 0 is a horizontal asymptote. 3. If the degree of p(x) is greater than the degree of q(x), then f (x) has no horizontal asymptote.
x1 . What happens when x = 1? What happens when x = 2? Graph this function for x+2 5 x 5 and 5 y 5.
Example. Let f (x) =
x f (x) -1.9
-1.99
-1.999
-1.9999
x f (x)
-2.1
-2.01
-2.001
-2.0001
Finding vertical asymptotes. Let f (x) =
look at places where the denominator q(x) = 0.
p(x) q(x)
be a rational function. To nd vertical asymptotes,
Math 107 Workbook
Sections 9.4 and 9.5 Rational Functions
83
For each of the following rational functions, nd all horizontal and vertical asymptotes (if there are any), all x-intercepts (if there are any), and the y-intercept. Find exact and approximate values when possible. Then, give a rough sketch of the function. 1. f (x) = 3x 4 7x + 1
2. f (x) =
x2 + 10x + 24 x2 2x + 1
3. f (x) =
2x3 + 1 x2 + x
4. f (x) =
(x2 4)(x2 + 1) x6
5. f (x) =
2x + 1 6x2 + 31x 11
Math 107 Workbook 6. f (x) = 2x c , where c and d are constants, and c = 0. (x c)(3x + d)
2
84
7. f (x) =
1 1 + x3 x5 Hint: First, nd a common denominator.
8. f (x) =
x5 2x4 9x + 18 8x3 + 2x2 3x 5 Hint: x 2x4 9x + 18 = x4 (x 2) 9(x 2)
9. f (x) =
2 1 + +3 x1 x+2 Hint: First, nd a common denominator.

**Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.**

Below is a small sample set of documents:

Sonoma - PHILOSOPHY - 101

In Search of Harmony: Becoming Your Own Hero at Work Copyright 2002 by Vivian Ellis Zabriskie Chapter 9 Resiliency Strategies: Coping with Stress/Avoiding DespairHey, this little kid gets roller skates, She puts them on, She stands up and almost f

Sonoma - PHILOSOPHY - 101

BUILDING CONSENSUS AROUND DAY LABOR ISSUES AGREEMENTS It will be the priority of the day labor center to help facilitate the safe, just employment of day laborers in Graton at one hiring site. To this end the members of the Representative Stakeholder

Sonoma - PHILOSOPHY - 101

Reading more international newspapers, such as the French Figaro and the British Guardian, has me made less lonely.1 I feel more connected to other people who have thoughts similar to mine .| | | | | V Look for the footnote at the bottom of this pa

Sonoma - PHILOSOPHY - 101

Reading more broadly, especially the novels of Thomas Mann, has me made more at easy with my own strange ideas.1 I feel more connected to other people who followed their passion .| | | | | V Look for the footnote at the bottom of this page (on in P

Sonoma - PSY - 445

Psychology 445: Advanced Research Methods Below is the grading scheme that I will use for your final papers (although absolute point value I will leave to my discretion.) Spelling, grammar and general flow of the paper do count. You can work in partn

Sonoma - PSY - 445

Psychology 445L Spring 2008 Homework #2 Name: _ Date: _ In all problems involving any figuring, please show your work. 1. A researcher studied the relation between psychotherapists degree of empathy and their patients satisfaction with therapy. As a

Sonoma - PSY - 445

Psychology 445L Spring 2008 Homework #1 Name: _ Date: _ Although most real-life statistics problems are solved using computer programs, please do these problems by hand as one method to ingrain the method in your mind. In all problems involving any f

Sonoma - PSY - 445

Psychology 445L Spring 2008 Homework #3 Name: _ Date: _ In all problems involving any figuring, please show your work. 1. A sports psychologist working with hockey players found that knowledge of physiology correlates .4 with number of injuries recei

Sonoma - P - 114

Sonoma State UniversityPhysics 114 J.S. TennSummary of Unit Systems in MechanicsQuantity mass length time velocity Dening equation Dimension M L T SI unit kg m s m/s cgs unit g cm s cm/s relationship 1 kg = 10 g 1 m = 10 cm2 3Other units slug

Temple - MATH - 067

Math C067 - Confidence IntervalsRichard Beigel April 10, 2006Parameters vs. Statistics A parameter is a numerical property (characteristic) of a population. Examples: (population mean) 2 (population variance) (population standard deviation) A

Temple - MATH - 067

Math C067 Bayes TheoremRichard Beigel February 20, 2006Gold and Silver Balls. Bill has three boxes. Their contents are Box 1: Two silver balls Box 2: One silver ball and one gold ball Box 3: Two gold balls. Carolyn picks one of the boxes at random

Temple - CIS - 601

Computer VisionMatlabA good choice for vision program development because Easy to do very rapid prototyping Quick to learn, and good documentation A good library of image processing functions Excellent display capabilities Widely used for teac

Temple - CIS - 617

Solutions by Archana GuptaQuestion 1 (Chapter 3: 10) An 8-bit byte with binary value 10101111 is to be encoded using an even-parity Hamming code. What is the binary value after encoding? Answer Check bits are inserted at positions that are powers of

Temple - CIS - 601

Computer VisionLecture 9: Stereoscopic visionLast lecture Layout, camera coordinates, slant and tilt The role of layout representations The geometry of image formation: perspective projection Monocular methods for depthThis lecture Seeing wi

Temple - CIS - 601

Computer VisionLecture 8: Introduction to 3-D VisionSo far we have looked at methods for analysing 2-D images. Vision is concerned with obtaining the information needed to interact successfully with a 3-D environment. Layout The layout of a scene m

Temple - CIS - 617

Archana Gupta Question 1 Consider building a CSMA/CD network running at 1 Gbps over a 1 km cable with no repeaters. The signal speed in the cable is 200000km/sec. What is the minimum frame size? Answer We must keep in mind that in CSMA/CD , for a sta

Temple - CIS - 166

Section 7.4 Closures of RelationsDefinition: The closure of a relation R with respect to property P is the relation obtained by adding the minimum number of ordered pairs to R to obtain property P. In terms of the digraph representation of R To fi

Temple - CIS - 166

Section 8.3 Representing Graphs and Graph Isomorphism We wish to be able to determine when two graphs are identical except perhaps for the labeling of the vertices. We derive some alternate representations which are extensions of connection matrices

Temple - CIS - 166

Section 7.5 Equivalence Relations Now we group properties of relations together to define new types of important relations. _ Definition: A relation R on a set A is an equivalence relation iff R is reflexive symmetric and transitive __ It is easy

Temple - CIS - 166

Section 7.1 Relations and Their PropertiesDefinition: A binary relation R from a set A to a set B is a subset R A B. Note: there are no constraints on relations as there are on functions. We have a common graphical representation of relations: De

Temple - CIS - 166

Section 8.2 Graph Terminology Undirected Graphs Definition: Two vertices u, v in V are adjacent or neighbors if there is an edge e between u and v. The edge e connects u and v. The vertices u and v are endpoints of e. _ Definition: The degree of a ve

Temple - CHAPTERS - 1100

Temple - CHAPTERS - 1800

1Women Must Occupy and Give Themselves the Place They DeserveWomens Activism and Feminist Agency in Mozambique and NicaraguaMany women in the countryside challenge the way things are without using the word gender, without ever studying women and m

Temple - CHAPTERS - 1400

INTRODUCTIONhen Andy Warhol began making films in 1963, blow jobs, although widely given and received, to be sure, were still a taboo subject. Any claim that, today, this taboo has been lifted, is a misjudgment, despite the sexual liberation of the

Temple - CIS - 339

Joe Mohen and Julia Glidden The March 2000 Arizona Democratic Partys presidential preference primary illustrates the promise of accessibility, inclusiveness, and accountability in online elections.The case forInternet VotingHow the Internet m

Temple - BA - 951

2006Mehri21The Darker Side of Lean:An Insiders Perspective on the Realities of the Toyota Production SystemDarius Mehri*Executive OverviewThe Toyota Production System (TPS) has been lauded as the pinnacle of flexible, just-in-time manufact

Temple - BA - 951

BA 951 IMBA Section Spring 2007Professor Ram Mudambi Ram.Mudambi@temple.eduThis is the capstone business administration course of your program and is intended to fulfill two objectives. First, it will provide a framework to the issues and decisio

Temple - BA - 951

OTHER PEOPLES MONEY Synopsis New England Wire & Cable (NECW) is a Rhode Island manufacturer with multiple divisions. The flagship cable and wire unit is unprofitable; all the other divisions are profitable. The company is debt-free, with a fully fund

Temple - BA - 804

Article 14The Sacking of ArgentinaTHE IMF DESERVES TO BE BLAMED, BUT SO DOES THE COUNTRY'S WILLING POLITICAL CLASS. TIM FRASCABuenos AiresIn December of last year, Argentina's decade-long and highly celebrated experiment as the poster child of m

Sonoma - BIO - 121

Figure 41.17 Enzymatic digestion in the human digestive systemDigestion Carbohydrate digestion starts in the mouth Protein digestion starts in the stomach Nucleic acids & fats start in the small intestine Everything completely digested and abso

Sonoma - BIO - 121

Updates Taxon Factsheets Due Next Week in Lab Instructions are posted on lab webpageFigure 34.7 Phylogeny of the major groups of extant vertebrates On Your Own Primates & the Evolution of Homo sapiens Pp. 707-715 Revised Lecture Syllabus

Sonoma - MUSIC - 270

Music 270 Assignment #1 Reading and Playing Music: A Different Approach On the two pages following these instructions is a copy of the score for Doodling, the piano piece I played in class on 2/1/07. After I performed Doodling, I pointed out that wha

Sonoma - MUSIC - 300

Durand 1 Christa Durand MUS 300 May Copland Aaron Copland was born to Sarah Mittenthal and Harris Copland on November 14 in the year 1900. One of his earliest musical memories was refraining from going out side to play, to stay inside and listen to h

Sonoma - MUSIC - 270

Music 270MUSICKINGWHAT IT TAKES TO MUSICK DO IT (just play until it is time to stop) THEN WHAT? DO IT (again) GOING FURTHER LIMITS FOR HOW LONG? (note recorded time is limited) HOW MANY EVENTS? (just one, or ) HOW MANY PLAYERS? RANGE (low to high

Sonoma - MUSIC - 300

Roth-Newell 1 Bennett Roth-Newell MUS 300: Music of Charles IvesThe notorious American classical composers have all found a way to create their own unique and distinguished style that creates separation as well as variety in the music and among the

Sonoma - MUSIC - 270

270 GRADEBOOKSTUDENT ID points possible 001103393 001321962 001513452 001661925 001752990 001784918 001786049 001797697 001821110 001831718 001895652 001918922 001929803 001957259 001993399 001995063 001995908 001997520 001998586 002007387 002011937

Sonoma - MUSIC - 301

MUSIC 301 Grade RosterSTUDENT ID NUMBER max points => 0110661 0112650 0136648 0156629 0332727 0617401 1007830 1090380 1102912 1105096 1123322 1138155 1159982 1173931 1177844 1275357 1300941 1338914 1364615 1365551 1435855 1439196 1463064 1474842 148

Temple - C - 067

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C067.001. Course Title: Elements of Statistics. Time: MWF 9:40-10:30. Place: BB 205. Instructor: Richard Beigel. Instructo

Temple - MATH - 0271

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0271.001. Course Title: Modern Geometry. Time: Arranged (One hour per week). Place: Arranged. Instructor: Bruce P. Conrad.

Temple - C - 011

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C085.011. Course Title: Calculus I. Time: MWF 2:00pm - 3:10pm. Place: BB 407. Instructor: Whondel Sterling. Instructor Off

Temple - C - 073

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C073.002. Course Title: College Algebra. Time: MWF 10:00-11:10. Place: AC00004. Instructor: Nahed Hamid. Instructor Oce: W

Temple - C - 101

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C085.101. Course Title: Calculus I. Time: TR, 5:30-7:10 p.m. Place: Ambler Widener Hall 214. Instructor: Ruth Trubnik. Ins

Temple - MATH - 0351

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0351.001. Course Title: Partial Dierrential Equations. Time: Mondays, Wednesdays, Fridays: 11:40-12:30. Place: Barton Hall

Temple - C - 073

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C073.001. Course Title: Math C073 COLLEGE ALGEBRA. Time: MWF: 8:40 AM - 9:50 AM. Place: Anderson Hall, Room #AC-4. Instruc

Temple - C - 085

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C085.003. Course Title: Mathematics C085.003. Time: 11:20 AM - 12:30 PM. Place: Barton Hall BB403. Instructor: David R. Hi

Temple - C - 010

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C074.010. Course Title: Math C074 Precalculus. Time: TR 2:10pm - 4:00pm. Place: BB109. Instructor: Marilena Downing. Instr

Temple - MATH - 0086

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0086.003. Course Title: Calculus II. Time: MWF 11:20 AM - 12:30 PM. Place: BARTON HALL CLASSROOMS 405. Instructor: Scott A

Temple - C - 065

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C065.008. Course Title: Elements of Math Thought, MATH 65. Time: 8:40AM-9:30AM, MWF. Place: BB101. Instructor: Whondel Ste

Temple - C - 073

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C073.004. Course Title: College Algebra. Time: MWF 12:40-1:50. Place: BB00205. Instructor: Nahed Hamid. Instructor Oce: Wa

Temple - MATH - 0045

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0045.003. Course Title: Elementary Algebra - Mathematics 0045.003. Time: MWF 11:40 a.m. to 12:30 p.m. and Recitation on Tu

Temple - C - 074

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C074.002. Course Title: Math C074.002 Precalculus. Time: 10:00 - 11:10 - Monday, Wednesday, Friday. Place: AC00007. Instru

Temple - MATH - 0086

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0086.009. Course Title: Calculus II. Time: TR 5:40-7:30 p.m. Place: 4xx Barton Hall Classroom Building. Instructor: Louis

Temple - C - 077

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C077.003. Course Title: C077 Differential and Integral Calculus. Time: MWF 12:40-1:50pm. Place: Barton Hall Classrooms 401

Temple - C - 085

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C085.006. Course Title: Calculus I. Time: TR 10:10-12:00. Place: AC 04. Instructor: Georgia Triantallou. Instructor Oce: W

Temple - C - 055

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C055.019. Course Title: College Math-C055-Section # 019. Time: T.TH. : 2:40PM-4:00PM. Place: BB205. Instructor: Prakash C.

Temple - C - 055

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C055.020. Course Title: College Math. Time: TR 11:40 - 1:00. Place: BB202. Instructor: Raymond F. Coughlin. Instructor Oce

Temple - MATH - 0127

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0127.001. Course Title: Calculus III. Time: MWF 11:20-12:30. Place: BB 407. Instructor: Farzana Chaudhry. Instructor Oce:

Temple - C - 073

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C073.003. Course Title: College Algebra. Time: MWF 11:20-12:30. Place: BB00401. Instructor: Nahed Hamid. Instructor Office

Temple - C - 074

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics C074.007. Course Title: Math C074 Precalculus. Time: TR 10:10am - 12:00pm. Place: BB103. Instructor: Marilena Downing. Ins

Temple - MATH - 0127

College of Science and TechnologyDepartment of MathematicsTEMPLE UNIVERSITYSpring 2007Course SyllabusCourse: Mathematics 0127.000. Course Title: Calculus III. Time: MWF 8:40-10:00. Place: XYZ001. Instructor: Boris A. Datskovsky. Instructor Oce

Temple - CHAPTERS - 1400

If God had found a reason to take a snapshot of Paradise, it would have shown Main Street to be the trunk of an evergreen, roads sprouting like boughs so ragged and droopy the whole thing resembled a Christmas tree left by the curb. Once, fifty resor