Pre-Calculus
Know-It-ALL
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About the Author
Stan Gibilisco is an electronics engineer, researcher, and mathematician
who has authored a number of titles for the McGraw-Hill Demystified series,
along with more than 30 other books and dozens of magazine articles. His
work has been published in several languages.
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Pre-Calculus
Know-It-ALL
Stan Gibilisco
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To Emma, Samuel, Tony, and Tim
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Contents
Preface
xi
Acknowledgments xiii
Part 1 Coordinates and Vectors 1
1 Cartesian Two-Space 3
How It’s Assembled 3
Distance of a Point from Origin 8
Distance between Any Two Points 12
Finding the Midpoint 15
Practice Exercises 18
2 A Fresh Look at Trigonometry
Circles in the Cartesian Plane 21
Primary Circular Functions 23
Secondary Circular Functions 30
Pythagorean Extras 33
Practice Exercises 36
21
3 Polar Two-Space 37
The Variables 37
Three Basic Graphs 40
Coordinate Transformations
Practice Exercises 52
45
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viii
Contents
4 Vector Basics 55
The “Cartesian Way” 55
The “Polar Way” 62
Practice Exercises 71
5 Vector Multiplication 73
Product of Scalar and Vector 73
Dot Product of Two Vectors 79
Cross Product of Two Vectors 82
Practice Exercises 88
6 Complex Numbers and Vectors 90
Numbers with Two Parts 90
How Complex Numbers Behave 95
Complex Vectors 101
Practice Exercises 109
7 Cartesian Three-Space 111
How It’s Assembled 111
Distance of Point from Origin 116
Distance between Any Two Points 120
Finding the Midpoint 122
Practice Exercises 126
8 Vectors in Cartesian Three-Space
How They’re Defined 128
Sum and Difference 134
Some Basic Properties 138
Dot Product 141
Cross Product 144
Some More Vector Laws 146
Practice Exercises 150
128
9 Alternative Three-Space 152
Cylindrical Coordinates 152
Cylindrical Conversions 156
Spherical Coordinates 159
Spherical Conversions 164
Practice Exercises 171
10 Review Questions and Answers
Part 2
Analytic Geometry
172
209
11 Relations in Two-Space 211
What’s a Two-Space Relation? 211
What’s a Two-Space Function? 216
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Contents
Algebra with Functions 222
Practice Exercises 227
12 Inverse Relations in Two-Space 229
Finding an Inverse Relation 229
Finding an Inverse Function 238
Practice Exercises 247
13 Conic Sections 249
Geometry 249
Basic Parameters 253
Standard Equations 258
Practice Exercises 263
14 Exponential and Logarithmic Curves
266
Graphs Involving Exponential Functions 266
Graphs Involving Logarithmic Functions 273
Logarithmic Coordinate Planes 279
Practice Exercises 283
15 Trigonometric Curves
285
Graphs Involving the Sine and Cosine 285
Graphs Involving the Secant and Cosecant 290
Graphs Involving the Tangent and Cotangent 296
Practice Exercises 302
16 Parametric Equations in Two-Space
What’s a Parameter? 304
From Equations to Graph 308
From Graph to Equations 314
Practice Exercises 318
17 Surfaces in Three-Space
Planes 320
Spheres 324
Distorted Spheres 328
Other Surfaces 337
Practice Exercises 343
304
320
18 Lines and Curves in Three-Space
Straight Lines 345
Parabolas 350
Circles 357
Circular Helixes 363
Practice Exercises 370
345
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ix
x
Contents
19 Sequences, Series, and Limits 373
Repeated Addition 373
Repeated Multiplication 378
Limit of a Sequence 382
Summation “Shorthand” 385
Limit of a Series 388
Limits of Functions 390
Memorable Limits of Series 394
Practice Exercises 396
20 Review Questions and Answers
Final Exam
399
436
Appendix A Worked-Out Solutions to Exercises: Chapter 1-9
466
Appendix B Worked-Out Solutions to Exercises: Chapter 11-19
Appendix C
Answers to Final Exam Questions
Appendix D
Special Characters in Order of Appearance
Suggested Additional Reading
Index
581
583
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579
514
Preface
This book is intended to complement standard pre-calculus texts at the high-school, trade-school,
and college undergraduate levels. It can also serve as a self-teaching or home-schooling supplement.
Prerequisites include beginning and intermediate algebra, geometry, and trigonometry. Pre-Calculus
Know-It-ALL forms an ideal “bridge” between Algebra Know-It-ALL and Calculus Know-It-ALL.
This course is split into two major sections. Part 1 (Chapters 1 through 10) deals with coordinate systems and vectors. Part 2 (Chapters 11 through 20) is devoted to analytic geometry. Chapters
1 through 9 and 11 through 19 end with practice exercises. They’re “open-book” quizzes. You may
(and should) refer to the text as you work out your answers. Detailed solutions appear in Appendices A and B. In many cases, these solutions don’t represent the only way a problem can be figured
out. Feel free to try alternatives!
Chapters 10 and 20 contain question-and-answer sets that finish up Parts 1 and 2, respectively.
These chapters aren’t tests. They’re designed to help you review the material, and to strengthen your
grasp of the concepts.
A multiple-choice Final Exam concludes the course. It’s a “closed-book” test. Don’t look back
at the chapters, or use any other external references, while taking it. You’ll find these questions more
general (and easier) than the practice exercises at the ends of the chapters. The exam is meant to
gauge your overall understanding of the concepts, not to measure how fast you can perform calculations or how well you can memorize formulas. The correct answers are listed in Appendix C.
I’ve tried to introduce “mathematicalese” as the book proceeds. That way, you’ll get used to the
jargon as you work your way through the examples and problems. If you complete one chapter a
week, you’ll get through this course in a school year with time to spare, but don’t hurry. Proceed at
your own pace.
Stan Gibilisco
xi
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Acknowledgments
I extend thanks to my nephew Tony Boutelle, a technical writer based in
Minneapolis, Minnesota, who offered insights and suggestions from the
viewpoint of the intended audience, and found a few arithmetic errors before they got into print!
I’m also grateful to Andrew A. Fedor, M.B.A., P.Eng (), a
freelance consultant from Hampton, Ontario, Canada, for his proofreading
help. Andrew has often provided suggestions for my existing publications
and ideas for new ones.
xiii
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Pre-Calculus
Know-It-ALL
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PART
1
Coordinates and Vectors
1
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CHAPTER
1
Cartesian Two-Space
If you’ve taken a course in algebra or geometry, you’ve learned about the graphing system
called Cartesian (pronounced “car-TEE-zhun”) two-space, also known as Cartesian coordinates
or the Cartesian plane. Let’s review the basics of this system, and then we’ll learn how to calculate distances in it.
How It’s Assembled
We can put together a Cartesian plane by positioning two identical real-number lines so they
intersect at their zero points and are perpendicular to each other. The point of intersection is
called the origin. Each number line forms an axis that can represent the values of a mathematical variable.
The variables
Figure 1-1 shows a simple set of Cartesian coordinates. One variable is portrayed along a horizontal line, and the other variable is portrayed along a vertical line. The number-line scales are
graduated in increments of the same size.
Figure 1-2 shows how several ordered pairs of the form (x,y) are plotted as points on the
Cartesian plane. Here, x represents the independent variable (the “input”), and y represents
the dependent variable (the “output”). Technically, when we work in the Cartesian plane, the
numbers in an ordered pair represent the coordinates of a point on the plane. People sometimes
say or write things as if the ordered pair actually is the point, but technically the ordered pair
is the name of the point.
Interval notation
In pre-calculus and calculus, we’ll often want to express a continuous span of values that a
variable can attain. Such a span is called an interval. An interval always has a certain minimum
value and a certain maximum value. These are the extremes of the interval. Let’s be sure that
3
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4
Cartesian Two-Space
Positive
dependentvariable
axis
6
Negative
independentvariable
axis
4
2
–6
–4
–2
2
6
4
–2
Positive
independentvariable
axis
–4
Negative
dependent–6
variable
axis
Figure 1-1
The Cartesian plane consists of two realnumber lines intersecting at a right angle,
forming axes for the variables.
y
6
(–4, 5)
Origin = (0, 0)
4
(4, 3)
2
x
–6
–4
–2
2
4
6
–2
(–5, –3)
–4
–6
Figure 1-2
Ordered pairs
are of
the form (x, y)
(1, –6)
Five ordered pairs (including the origin)
plotted as points on the Cartesian plane. The
dashed lines are for axis location reference.
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How It’s Assembled
5
you’re familiar with standard interval terminology and notation, so it won’t confuse you later
on. Consider these four situations:
0
−1 ≤ y < 0
4
−p ≤ q ≤ p
These expressions have the following meanings, in order:
•
•
•
•
The value of x is larger than 0, but smaller than 2.
The value of y is larger than or equal to −1, but smaller than 0.
The value of z is larger than 4, but smaller than or equal to 8.
The value of q is larger than or equal to −p, but smaller than or equal to p.
The first case is an example of an open interval, which we can write as
x ∈ (0,2)
which translates to “x is an element of the open interval (0,2).” Don’t mistake this open
interval for an ordered pair! The notations look the same, but the meanings are completely
different. The second and third cases are examples of half-open intervals. We denote this type
of interval with a square bracket on the side of the included value and a rounded parenthesis
on the side of the non-included value. We can write
y ∈ [−1,0)
which means “y is an element of the half-open interval [−1,0),” and
z ∈ (4,8]
which means “z is an element of the half-open interval (4,8].” The fourth case is an example of
a closed interval. We use square brackets on both sides to show that both extremes are included.
We can write this as
q ∈ [−p,p]
which translates to “q is an element of the closed interval [−p,p].”
Relations and functions
Do you remember the definitions of the terms relation and function from your algebra courses?
(If you read Algebra Know-It-All, you should!) These terms are used often in pre-calculus, so
it’s important that you be familiar with them. A relation is an operation that transforms, or
maps, values of a variable into values of another variable. A function is a relation in which there
is never more than one value of the dependent variable for any value of the independent variable. In other words, there can’t be more than one output for any input. (If a particular input
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6
Cartesian Two-Space
produces no output, that’s okay.) The Cartesian plane gives us an excellent way to illustrate
relations and functions.
The axes
In a Cartesian plane, both axes are linear, and both axes are graduated in increments of the
same size. On either axis, the change in value is always directly proportional to the physical
displacement. For example, if we travel 5 millimeters along an axis and the value changes by
1 unit, then that fact is true everywhere along that axis, and it’s also true everywhere along the
other axis.
The quadrants
Any pair of intersecting lines divides a plane into four parts. In the Cartesian system, these
parts are called quadrants, as shown in Fig. 1-3:
• In the first quadrant, both variables are positive.
• In the second quadrant, the independent variable is negative and the dependent variable
is positive.
• In the third quadrant, both variables are negative.
• In the fourth quadrant, the independent variable is positive and the dependent variable
is negative.
y
6
II
I
4
Second
quadrant
2
First
quadrant
x
–6
–4
–2
Third
quadrant
III
2
–2
4
6
Fourth
quadrant
–4
IV
–6
Figure 1-3
The Cartesian plane is divided into
quadrants. The first, second, third, and
fourth quadrants are sometimes labeled I, II,
III, and IV, respectively.
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How It’s Assembled
The quadrants are sometimes labeled with Roman numerals, so that
•
•
•
•
Quadrant I is at the upper right
Quadrant II is at the upper left
Quadrant III is at the lower left
Quadrant IV is at the lower right
If a point lies on one of the axes or at the origin, then it is not in any quadrant.
Are you confused?
Why do we insist that the increments be the same size on both axes in a Cartesian two-space
graph? The answer is simple: That’s how the Cartesian plane is defined! But there are other
types of coordinate systems in which this exactness is not required. In a more generalized
system called rectangular coordinates or the rectangular coordinate plane, the two axes can be
graduated in divisions of different size. For example, the value on one axis might change by
1 unit for every 5 millimeters, while the value on the other axis changes by 1 unit for every
10 millimeters.
Here’s a challenge!
Imagine an ordered pair (x,y), where both variables are nonzero real numbers. Suppose that you’ve
plotted a point (call it P) on the Cartesian plane. Because x ≠ 0 and y ≠ 0, the point P does not lie
on either axis. What will happen to the location of P if you multiply x by −1 and leave y the same?
If you multiply y by −1 and leave x the same? If you multiply both x and y by −1?
Solution
If you multiply x by −1 and do not change the value of y, P will move to the opposite side of the
y axis, but will stay the same distance away from that axis. The point will, in effect, be “reflected”
by the y axis, moving to the left if x is positive to begin with, and to the right if x is negative to
begin with.
•
•
•
•
If P starts out in the first quadrant, it will move to the second.
If P starts out in the second quadrant, it will move to the first.
If P starts out in the third quadrant, it will move to the fourth.
If P starts out in the fourth quadrant, it will move to the third.
If you multiply y by −1 and leave x unchanged, P will move to the opposite side of the x axis, but
will stay the same distance away from that axis. In a sense, P will be “reflected” by the x axis, moving straight downward if y is initially positive and straight upward if y is initially negative.
•
•
•
•
If P starts out in the first quadrant, it will move to the fourth.
If P starts out in the second quadrant, it will move to the third.
If P starts out in the third quadrant, it will move to the second.
If P starts out in the fourth quadrant, it will move to the first.
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8
Cartesian Two-Space
If you multiply both x and y by −1, P will move diagonally to the opposite quadrant. It will, in
effect, be “reflected” by both axes.
•
•
•
•
If P starts out in the first quadrant, it will move to the third.
If P starts out in the second quadrant, it will move to the fourth.
If P starts out in the third quadrant, it will move to the first.
If P starts out in the fourth quadrant, it will move to the second.
If you have trouble envisioning these point maneuvers, draw a Cartesian plane on a piece of graph
paper. Then plot a point or two in each quadrant. Calculate how the x and y values change when you
multiply either or both of them by −1, and then plot the new points.
Distance of a Point from Origin
On a straight number line, the distance of any point from the origin is equal to the absolute
value of the number corresponding to the point. In the Cartesian plane, the distance of a
point from the origin depends on both of the numbers in the point’s ordered pair.
An example
Figure 1-4 shows the point (4,3) plotted in the Cartesian plane. Suppose that we want to find
the distance d of (4,3) from the origin (0,0). How can this be done?
We can calculate d using the Pythagorean theorem from geometry. In case you’ve forgotten
that principle, here’s a refresher. Suppose we have a right triangle defined by points P, Q, and
R. Suppose the sides of the triangle have lengths b, h, and d as shown in Fig. 1-5. Then
b2 + h2 = d 2
We can rewrite this as
d = (b 2 + h 2)1/2
where the 1/2 power represents the nonnegative square root. Now let’s make the following
point assignments between the situations of Figs. 1-4 and 1-5:
• The origin in Fig. 1-4 corresponds to the point Q in Fig. 1-5.
• The point (4,0) in Fig. 1-4 corresponds to the point R in Fig. 1-5.
• The point (4,3) in Fig. 1-4 corresponds to the point P in Fig. 1-5.
Continuing with this analogy, we can see the following facts:
• The line segment connecting the origin and (4,0) has length b = 4.
• The line segment connecting (4,0) and (4,3) has height h = 3.
• The line segment connecting the origin and (4,3) has length d (unknown).
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