Precalculus Study Guide
Copyright Maplesoft, a division of Waterloo Maple Inc., 2024
Contents
Introduction
The eleven chapters of this Precalculus Study Guide contain concepts and computational techniques that support the study of calculus of a single variable. The concepts can be met in a course specifically called Precalculus, or even in a course called College Algebra. Yet, many of these ideas are rooted in the first and second algebra courses met in high school.
This Study Guide should serve both the student looking for a focused review, and the student who wishes for some deeper insight into material being met for a first time. It is designed to teach mastery of precalculus mathematics, ultimately through Maple.
Each of its eleven chapters begins with a short introduction that sketches the important ideas of the chapter. Within each introduction is a glossary of terms met in the chapter, terms that are hyperlinked to the Maple Mathematics Dictionary. The first time one of these terms is encountered in each subsection, it is hyperlinked to the dictionary.
After the introduction, each chapter presents a selection of typical problems that span the material of the chapter. A list of such problems is then followed by a section of solutions. Each problem is solved in four ways. First, there is a mathematical solution, one delivered the way it would appear in any standard textbook. These solutions do not use any technology other than an occasional computer-drawn graph. Essentially, these are "by hand" solutions that use classical techniques appearing in traditional texts.
Following these "mathematical" solutions, there are Maplet solutions. Maple's Maplet technology provides for custom-designed user interfaces called Tutors. Built into Maple are nearly 50 such graphical user interfaces to a wide spectrum of Maple functionality. This study guide provides 21 additional tutors designed specifically around the typical problems contained therein.
These tutors provide a "point-and-click" approach to the solutions. Clearly, they hide some of the details of the calculations, but certainly they allow the user to focus on the key ideas, free of the need to learn either by-hand manipulations, or Maple syntax. These tutors allow a student to obtain all the key features of a solution with just a few clicks of the mouse.
Maple's ease-of-use features provide a "syntax-free" paradigm that complements both the Maplet and Programmatic solutions. These Interactive Solutions are presented in a two-column format, the left-hand column containing the stage directions for Maple interactions; the right-hand, space to implement these scripted calculations.
Finally, in the "Programmatic Solution" sections each of the typical problems is solved with an appropriate set of annotated Maple commands. In these solutions, Maple syntax is freely used (and explained) so that the user who has studied the mathematical, Maplet, and interactive solutions will see how to orchestrate the appropriate Maple commands to play the same symphony just heard thrice before.
The Table of Contents lists the tutors used in each chapter, and provides hyperlinks both to the worksheets constituting each chapter, and to the tutors relevant to each chapter.
Finally, an index collects all defined terms in one alphabetical list, and points to the chapters where these terms are used.
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List of Chapters
Chapter 1: Solving Inequalities
Inequality Tutor #1
Inequality Tutor #2
Chapter 2: Graphing Lines in the Cartesian Plane
Lines Tutor
Lines Tutor #2
Lines Tutor #3
Chapter 3: Graphing Polynomial Functions
Polynomial Tutor #1
Polynomial Tutor #2
Chapter 4: Graphing Rational Functions
Rational Function Tutor
Chapter 5: The Arithmetic of Functions
Arithmetic of Functions Tutor
Chapter 6: Composition of Functions
Composition Tutor
Chapter 7: Inverse Functions
Inverse Function Tutor #1
Inverse Function Tutor #2
Inverse Function Tutor #3
Chapter 8: Roots and Rational Powers
Complex Arithmetic Tutor
Plot Builder
Chapter 9: The Elementary Transcendental Functions
Transcendental Function Tutor
Inverse Tutor
Hyperbolic Function Identities Tutor
Inverse Hyperbolic Function Tutor
Transcendental Equation Tutor
Chapter 10: Piecewise-Defined Functions
Piecewise Function Tutor
Removable Singularity Tutor
Chapter 11: Functions Defined Implicitly
Implicit Function Tutor
Index
a b c d e f g h i l m n o p q r s t u z
a
Chapter
============================================================
abscissa 1
absolute value 1, 8, 10
additive inverse 10
amplitude 9
analytic geometry 2
arc cosecant 9
arc cosine 9
arc cotangent 9
arc hyperbolic cosecant 9
arc hyperbolic cosine 9
arc hyperbolic cotangent 9
arc hyperbolic secant 9
arc hyperbolic sine 9
arc hyperbolic tangent 9
arc secant 9
arc sine 9
arc tangent 9
Argand diagram 8
argument 8
asymptote 4, 6
axis of symmetry 5
b
binomial 8
bisector 2
bounded 6
c
Cartesian plane 2, 8 coefficient 3
complete the square 3
complex conjugate 3, 8
complex number 7, 8, 9
complex plane 8
composition 6, 7
converge 9
coordinate 2, 3
cosecant 9
cosine 9
cotangent 9
Cramer's Rule 2
cube 8
cube root 8
cubic 3
d
DeMoivre's laws 8
denominator 10
discretization 11
discriminant 11
divisor 4, 5, 6, 7, 9, 10, 11
e
elementary function 9
Euclidean geometry 2
even 4, 8
explicit 11
exponent 7, 8
exponential 7, 9, 10
exponential function 9
exponentiate 7, 8, 9
extension 10
extraneous roots 7
f
factor 3
floating-point 7, 8, 9, 11
function 7
fundamental theorem of algebra 3
g
greatest integer function 10
h
hyperbola 4
hyperbolic cosecant 9
hyperbolic cosine 9
hyperbolic cotangent 9
hyperbolic function 9
hyperbolic secant 9
hyperbolic sine 9
hyperbolic tangent 9
hypotenuse 8
i
identity function 7
imaginary 8
imaginary axis 8
imaginary part 8
implicit 11
increasing 7
inequality 1, 10, 11
infinity 4
integer 3, 8
intercept 2, 3, 4, 5
intersection 1
interval 1, 5, 10, 11
inverse function 7, 9
invertible 7
irrational number 9
iteration 9
l
leading coefficient 3
length 2, 8
limit 4
line 2
line segment 2
logarithm 7
long division 3
m
magnitude 8
many-one 8
matrix 2
midpoint 2, 5, 9
modulus 8
monic polynomial 3
n
natural logarithm 9
negative 2, 4, 7, 8, 10
nonnegative 1, 6, 10, 11
numerator 10
o
odd 4
one-to-one 7, 8, 9
onto 7
open interval 1, 8
p
parabola 3, 5, 7, 11
parallel 2
perpendicular 2
polynomial 3, 4
positive 1, 4, 7, 8, 10
power 8
Pythagoras' theorem 8
q
quadratic equation 7, 9, 11
quadratic formula 3, 5, 7, 9, 11
quartic 3
quotient 4, 5, 8
r
range 5, 6
ratio 4
rational function 4
rational number 8, 9
real line 1, 10
real axis 8
real number 3, 4, 5, 6, 7, 8, 9, 10, 11
real part 8
reciprocal 2, 4
reflection 7, 9
remainder 4
root 8
s
secant 9
sequence 9, 11
sine 9, 10
singularity 10
slope 2
square root 8, 11
subset 1, 10
symmetry 7, 8
synthetic division 3
t
tangent 3, 9
transversal 2
u
union 1
z
zero 3, 5
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