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**Unformatted text preview: **Precalculus Workbook 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
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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
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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ??
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Section 1.1 – Functions and Function Notation
Deﬁnition. 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:
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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 woman’s distance from Webster as a function of time.
b. Sketch a graph of the woman’s distance from Groton as a function of time.
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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) 3−1
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!
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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 diﬀerent 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)
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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 ﬁxed 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 signiﬁcance 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 signiﬁcance 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?
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Section 1.4 – Formulas For Linear Functions
1. You need to rent a car for one day and to compare the charges of 3 diﬀerent 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, ﬁnd 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 ﬁgure to the right. Given that the slope of one of the lines is −2, ﬁnd 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 2x−3y = 2 that goes through the point (1, 1). (b) The line perpendicular to 2x − 3y = 2 that goes through the point (1, 1).
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Section 1.5 – Geometric Properties of Linear Functions
Example. Given below are the equations for ﬁve diﬀerent 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) = .
.
.
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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)
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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 ﬁnal answer as a single fraction.
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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 Raider’s organization makes is a function of the number of people, n, in attendance. If each ticket costs $30.00, ﬁnd the domain and range of this function. Sketch its graph.
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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
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Section 2.4 – Inverse Functions
1. Use the two functions shown below to ﬁll 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)
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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).
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Section 2.5 – Concavity Introduction
Deﬁnitions. 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 ﬁrst, 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).”
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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
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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
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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?
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Chapter 3 – Algebra Gateway: Exponents
Evaluate or simplify without a calculator. Write your ﬁnal answer in the provided blank. 1. 9−1/2 + √ 0.01 =
2.
(xy 3 )2 x0 y 5
=
a3 b−1 3. √ a5/2 b1/2
=
4.
(AB)4 A−1 B −2
=
5. 2b−1 (b2 + b) − 2
=
6.
M + M −1 1 + M −2
=
7. 3 3 t3 + 7(t9 )1/3
=
8.
2km3 + k 2 m km−1
=
Solve for x 9. 81x = 3 x=
10.
6 =2 3ax
x=
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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 :
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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, ﬁnd 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?
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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.
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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 deﬁnitely positive?
y y=pqx
(b) What of these constants are deﬁnitely between 0 and 1?
x y=cd
(c) Which two of these constants are deﬁnitely 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
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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.
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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
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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
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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
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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 = 6e−0.04t Q = 5(1.2)t Q = 10(0.91)t
Growth or Decay Rate Per Year Continuous Per Year
Math 107 Workbook
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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?
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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
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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 deﬁned? 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 on the x and y axes. Does f (x) have any vertical asymptotes? Any horizontal asymptotes?
y
x
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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) = e−x . 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 x−2 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 oﬃce 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 Deﬁnition 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)
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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 diﬀerent 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
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Section 5.5 – The Family of Quadratic Functions
1. For each of the following, complete the square in order to ﬁnd 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 ﬁnd the vertex of the function.
1
-1
2
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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 ﬁrst time that d(t) = 0? The second time?
(b) When does the tomato reach its maximum height? How high is the tomato’s maximum height?
Math 107 Workbook
Section 6.1 – Periodic Functions
46
Deﬁnition. 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 function’s maximum and minimum output values. 3. The amplitude is the vertical distance between the function’s 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)?
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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
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Section 6.2 – Introduction
Angle Measurement in Circles
• Angles start from the positive x-axis. • Counterclockwise deﬁned to be positive.
y
x
Deﬁnition. 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 deﬁned 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, ﬁll in the blank with an angle between 0◦ and 360◦, diﬀerent from the ﬁrst 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(θ + 360◦ ) = (b) sin(θ + 180 ) = (c) cos(180 − θ) = (d) sin(180◦ − θ) = (e) cos(360◦ − θ) = (f) sin(360◦ − θ) = (g) sin(90◦ − θ) =
◦ ◦
y
(a,b) 1
θ
x
4. Use your calculator to ﬁnd 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
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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 oﬀ by an angle θ. Find the degree and radian measure of θ.
8
θ 2
θ 4
4
2. In the pictures below, ﬁnd the length of the arc cut oﬀ by each angle.
2π/3
2
80ο
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, ﬁnd 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
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Section 6.4 Supplement – The Unit Circle
θ θ cos θ
0 0◦
30
π 6
◦
45
π 4
◦
60
π 3
◦
90
π 2
◦
120
2π 3
◦
135
3π 4
◦
150◦
5π 6
π 180◦
sin θ
θ θ cos θ
210
7π 6
◦
225
5π 4
◦
240
4π 3
◦
270
3π 2
◦
300
5π 3
◦
315
7π 4
◦
330
11π 6 ◦
2π 360◦
sin θ
1
-3 -2 -1 -1
1
2
3
4
5
6
7
8
9
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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) Eﬀect 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” reﬂects the graph of a sinusoidal function about its midline.
Math 107 Workbook
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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 2Π 3Π 4Π
Π
Π
-3
3
0.8
2
4
Π 7
-3
-0.8
3
10 8
1 6 Π -1 2 4
-3
2
11
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2. For each of the following, ﬁnd the amplitude, the period, the phase shift, and the midline. (a) y = 2 cos(πx +
2π 3 )
−1
(b) y = 3 − sin(2x − 7π)
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 ﬁnd 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
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Section 6.6 – Reference Angles Supplement
Deﬁnition. The reference angle associated with an angle θ is the acute angle (having positive measure) formed by the x-axis and the terminal side of the angle θ.
Key Fact. If θ is any angle and θ′ is the reference angle, then
sin θ′ cos θ′ tan θ′ csc θ′ sec θ′ cot θ′ = = = = = = ± sin θ ± cos θ ± csc θ ± sec θ
± tan θ
± cot θ,
where the correct sign must be chosen based on the quadrant of the angle θ.
Example. For each of the following angles, sketch the angle and ﬁnd the reference angle.
(1) θ = 300◦ (2) θ =
4π 3
(3) θ = 135◦
(4) θ =
7π 6
Math 107 Workbook
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Section 6.6 – Other Trigonometric Functions
3 1. Suppose that sin θ = − 4 and that 3π 2
≤ θ ≤ 2π. Find the exact values of cos θ and sec θ.
2. Suppose that csc θ =
x 2
and that θ lies in the 2nd quadrant. Find expressions for cos θ and tan θ 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 θ u
s w
Math 107 Workbook
Section 6.7 – Inverse Trigonometric Functions
Preliminary Idea.
sin(π/6) = 1/2 means the same thing as .
58
Deﬁnition.
1. sin−1 x is the angle between − π and 2 2. tan−1 x is the angle between − π and 2
π 2 π 2
whose sine is x. whose tangent is x.
3. cos−1 x is the angle between 0 and π whose cosine is x.
Note. “sin−1 x”, “cos−1 x”, and “tan−1 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. sin−1
√ 2 2
=
√ 3. tan−1 ( 3) = 4. sin−1 (−1) =
Question. How would we ﬁnd all solutions to the equation sin x =
1 2
that lie between 0 and 2π?
Math 107 Workbook
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x sin x 0 0 π/6 √ 1/2 π/4 √2/2 π/3 3/2 π/2 1
x
sin−1 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 π/6 3/3 π/4 1 √ π/3 3 π/2 undeﬁned
x
tan−1 x
2
R
1
Q P
O
2 1 1 2
1
2
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Section 6.7 – Solving Trigonometric Equations
1. Solve each of the following trigonometric equations, giving all solutions between 0 and 2π. Give exact answers whenever possible. (a) sin θ = −
√ 3 2
1 (b) tan θ = − 4
(c) cos θ =
1 2
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2. Find all solutions to 2 sin x cos x + cos x = 0 that lie between 0 and 2π. 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 2π. Then, use reference angles to ﬁnd 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 θ =
cos θ
=
tan θ
=
θ (r,0)
Warmup example. A kite ﬂyer wondered how high her kite was ﬂying. 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
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Section 7.1 – The Laws of Sines and Cosines
1. Two ﬁre stations are located 25 miles apart, at points A and B. There is a forest ﬁre at point C. If ∠CAB = 54◦ and ∠CBA = 58◦ , which ﬁre 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 Eiﬀel 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 Eiﬀel 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
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Section 7.2 – Using Trigonometric Identities
Directions. On a separate sheet of paper, simplify each of the following as indicated by the instructions. Prob...