Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

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Mary Jane Sterling. Pre-Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice)

Pre-Calculus: 1001 Practice Problems For Dummies® To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Pre-Calculus: 1001 Practice Problems For Dummies Cheat Sheet” in the Search box. Table of Contents

Guide

Pages

Introduction

What You’ll Find

How This Workbook Is Organized

Part 1: The Questions

Part 2: The Answers

Beyond the Book

Where to Go for Additional Help

The Questions

Getting Started with Algebra Basics

The Problems You’ll Work On

What to Watch Out For

Identifying Which System or Systems a Number Belongs To

Recognizing Properties of Number Systems

Simplifying Expressions with the Order of Operations

Graphing Inequalities

Using Graphing Formulas

Applying Graphing Formulas

Solving Some Equations and Inequalities

The Problems You’ll Work On

What to Watch Out For

Using Interval and Inequality Notation

Solving Linear Inequalities

Solving Quadratic Inequalities

Solving Absolute Value Inequalities

Working with Radicals and Fractional Notation

Performing Operations Using Fractional Exponents

Factoring Using Fractional Notation

Solving Radical Equations

Rationalizing Denominators

Function Basics

The Problems You’ll Work On

What to Watch Out For

Using Function Notation to Evaluate Function Values

Determining the Domain and Range of a Function

Recognizing Even Functions

Identifying Odd Functions

Ruling Out Even and Odd Functions

Recognizing One-to-One Functions from Given Relations

Identifying One-to-One Functions from Equations

Recognizing a Function’s Inverse

Determining a Function’s Inverse

Executing Operations on Functions

Performing Function Composition

Doing More Function Composition

Using the Difference Quotient

Graphing and Transforming Functions

The Problems You’ll Work On

What to Watch Out For

Functions and Their Inverses

Sketching Quadratic Functions from Their Equations

Writing Equations from Graphs of Parabolas

Investigating and Graphing Radical Functions

Investigating Absolute Value Functions

Investigating the Graphs of Polynomial Functions

Investigating Rational Functions

Transformation of Functions

Transforming Selected Points Using Functions

Sketching Graphs Using Basic Functions and Transformations

Sketching More Graphs Using Basic Functions and Transformations

Polynomials

The Problems You’ll Work On

What to Watch Out For

Using Factoring to Solve Quadratic Equations

Solving Quadratic Equations by Using the Quadratic Formula

Using Completing the Square to Solve Quadratic Equations

Solving Polynomial Equations for Intercepts

Using Factoring by Grouping to Solve Polynomial Equations

Applying Descartes’s Rule of Signs

Listing Possible Roots of a Polynomial Equation

Dividing Polynomials

Using Synthetic Division to Divide Polynomials

Checking for Roots of a Polynomial by Using Synthetic Division

Writing Polynomial Expressions from Given Roots

Writing Polynomial Expressions When Given Roots and a Point

Graphing Polynomials

Writing Equations from Graphs of Polynomials

Exponential and Logarithmic Functions

The Problems You’ll Work On

What to Watch Out For

Understanding Function Notation

Graphing Exponential Functions

Solving Exponential Equations

Using the Equivalence to Rewrite Expressions

Using the Equivalence to Rewrite Expressions

Rewriting Logarithmic Expressions

Rewriting Logs of Products and Quotients as Sums and Differences

Solving Logarithmic Equations

Applying Function Transformations to Log Functions

Applying Logarithms to Everyday Life

Trigonometry Basics

The Problems You’ll Work On

What to Watch Out For

Using Right Triangles to Determine Trig Functions

Solving Problems by Using Right Triangles and Their Functions

Working with Special Right Triangles

Changing Radians to Degrees

Changing Degrees to Radians

Finding Angle Measures (in Degrees) in Standard Position

Determining Angle Measures (in Radians) in Standard Position

Identifying Reference Angles

Determining Trig Functions by Using the Unit Circle

Calculating Trig Functions by Using Other Functions and Terminal Side Positions

Using the Arc Length Formula

Evaluating Inverse Functions

Solving Trig Equations for x in Degrees

Calculating Trig Equations for x in Radians

Graphing Trig Functions

The Problems You’ll Work On

What to Watch Out For

Recognizing Basic Trig Graphs

Graphing Sine and Cosine

Applying Function Transformations to Graphs of Trig Functions

Writing New Trig Functions Using Transformations

Graphing Tangent and Cotangent

Interpreting Transformations of Trig Functions

Graphing Secant and Cosecant

Interpreting Transformations from Function Rules

Getting Started with Trig Identities

The Problems You’ll Work On

What to Watch Out For

Proving Basic Trig Identities

Returning to Basic Sine and Cosine to Solve Identities

Using Multiplication by a Conjugate to Solve Identities

Solving Identities After Raising a Binomial to a Power

Solving Identities After Factoring out a Common Function

Solving Identities After Combining Fractions

Performing Algebraic Processes to Make Identities More Solvable

Continuing with Trig Identities

The Problems You’ll Work On

What to Watch Out For

Using Identities That Add or Subtract Angle Measures

Confirming Double-Angle Identities

Using Identities That Double the Size of the Angle

Confirming the Statements of Multiple-Angle Identities

Creating Half-Angle Identities from Double-Angle Identities

Creating a Half-Angle Identity for Tangent

Using Half-Angle Identities to Simplify Expressions

Creating Products of Trig Functions from Sums and Differences

Using Product-to-Sum Identities to Evaluate Expressions

Using Sum-to-Product Identities to Evaluate Expressions

Applying Power-Reducing Identities

Using Identities to Determine Values of Functions at Various Angles

Working through Identities Using Multiple Methods

Working with Triangles and Trigonometry

The Problems You’ll Work On

What to Watch Out For

Applying the Law of Sines to Find Sides

Utilizing the Law of Sines to Find Angles

Using the Law of Sines for Practical Applications

Investigating the Ambiguous Case of the Law of Sines

Determining All Angles and Sides of a Triangle

Finding Side Measures by Using the Law of Cosines

Using the Law of Cosines to Determine an Angle

Applying the Law of Cosines to Real-World Situations

Finding Areas of Triangles by Using the Sine

Applying the Trig Formula for Area of a Triangle

Using the Trig Formula for Area in Various Situations

Solving Area Problems Needing Additional Computations

Finding Areas of Triangles by Using Heron’s Formula

Applying Heron’s Formula

Practical Applications Using Heron’s Formula

Tackling Practical Applications by Using Triangular Formulas

Complex Numbers and Polar Coordinates

The Problems You’ll Work On

What to Watch Out For

Writing Powers of i in Their Simplest Form

Adding and Subtracting Complex Numbers

Multiplying Complex Numbers

Using Multiplication to Divide Complex Numbers

Solving Quadratic Equations with Complex Solutions

Graphing Complex Numbers

Identifying Points with Polar Coordinates

Identifying Points Whose Angles Have Negative Measures

Converting Polar to Rectangular Coordinates

Converting Rectangular to Polar Coordinates

Recognizing Polar Curves

Conic Sections

The Problems You’ll Work On

What to Watch Out For

Identifying Conics from Their Equations

Rewriting Conic Equations in Standard Form

Writing Equations for Circles

Determining Foci and Axes of Symmetry of Parabolas

Finding the Vertices and Directrixes of Parabolas

Writing Equations of Parabolas

Determining Centers and Foci of Ellipses

Writing Equations of Ellipses

Determining Asymptotes of Hyperbolas

Writing Equations of Hyperbolas

Changing Equation Format from Trig Functions to Algebraic

Changing Equation Format from Algebraic to Trig

Systems of Equations and Inequalities

The Problems You’ll Work On

What to Watch Out For

Using Substitution to Solve Systems of Linear Equations with Two Variables

Using Elimination to Solve Systems of Linear Equations with Two Variables

Solving Systems of Equations Involving Nonlinear Functions

Solving Systems of Linear Equations

Solving Systems of Linear Equations with Four Variables

Graphing Systems of Inequalities

Decomposition of Fractions

Operating on Matrices

Changing Matrices to the Echelon Form

Solving Systems of Equations Using Augmented Matrices

Solving Systems of Equations Using the Inverse of the Coefficient Matrix

Applying Cramer’s Rule to Solve Systems of Equations

Sequences and Series

The Problems You’ll Work On

What to Watch Out For

Finding Terms of Sequences

Determining Rules for Sequences

Working with Recursively Defined Sequences

Adding Terms in an Arithmetic Series

Summing Terms of a Series

Finding Rules and Summing Terms of a Series

Calculating the Sum of a Geometric Series

Determining Formulas and Finding Sums

Counting Items by Using Combinations

Constructing Pascal’s Triangle

Applying Pascal’s Triangle

Utilizing the Binomial Theorem

Introducing Limits and Continuity

The Problems You’ll Work On

What to Watch Out For

Determining Limits from Graphs

Determining One-Sided Limits

Determining Limits from Function Values

Determining Limits from Function Rules

Applying Laws of Limits

Investigating Continuity

The Answers

Answers

Index. A

B

C

D

E

F

G

H

I

K

L

M

N

O

P

Q

R

S

T

U

V

W

About the Author

Dedication

Author’s Acknowledgments

WILEY END USER LICENSE AGREEMENT

Отрывок из книги

Pre-calculus is a rather difficult topic to define or describe. There’s a little bit of this, a lot of that, and a smattering of something else. But you need the mathematics considered to be pre-calculus to proceed to what changed me into a math major: calculus! Yes, believe it or not, I started out as a biology major — inspired by my high school biology teacher. Then I got to the semester where I was taking invertebrate zoology, chemistry, and calculus. (Yes, all at the same time.) All of a sudden, there was a bright light! An awakening! “So this is what mathematics can be!” Haven’t turned back since. Calculus did it for me, and my great preparation for calculus made the adventure wonderful.

Pre-calculus contains a lot of algebra, some trigonometry, some geometry, and some analytic geometry. These topics all get tied together, mixed up, and realigned until out pops the mathematics you’ll use when working with calculus. I keep telling my calculus students that “calculus is 60 percent algebra.” Maybe my figures are off a bit, but believe me, you can’t succeed in calculus without a good background in algebra (and trigonometry). The geometry is very helpful, too.

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47. Find the distance between the points (6, −5) and (−4, 3).

48. Find the midpoint of the segment between the points (−5, 2) and (7, −8).

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