Homework Helpers: Calculus

HOMEWORK HELPERS

Calculus

DENISE SZECSEI

Copyright © 2007 by Denise Szecsei

All rights reserved under the Pan-American and International Copyright Conventions. This book may not be reproduced, in whole or in part, in any form or by any means electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system now known or hereafter invented, without written permission from the publisher, The Career Press.

HOMEWORK HELPERS: CALCULUS

EDITED BY JODI BRANDON

TYPESET BY EILEEN DOW MUNSON

Cover design by Lu Rossman/Digi Dog Design NYC

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Library of Congress Cataloging-in-Publication Data

Szecsei, Denise.\

Homework helpers. Calculus / by Denise Szecsei.

      p. cm.

Includes index.

ISBN-13: 978-1-56414-914-5

ISBN-10: 1-546414-914-5

1. Calculus. 2. Calculus—Problems, exercises, etc. I. Title.

Qa300.S985  2006

515—dc22

2006026398

Dedication

This book is dedicated to Mickey Perry, the first of my guides through the 9 circles of mathematics.

Acknowledgments

This book was a group effort, and I would like to thank the people who helped throughout the entire production.

I would like to thank Michael Pye, Kristen Parkes, and everyone else at Career Press who worked on this project. Jessica Faust handled the logistics so that I could focus on writing.

I am grateful for Kendelyn Michaels’s willingness to assist in the development of this book. I benefited greatly from her review of the manuscript and her suggestions for improvements. I appreciate her continued participation in my writing projects.

Alic Szecsei helped reduce the number of typographical errors in the manuscript and was willing to spend many hours working out the solutions to the review problems.

I want to thank my family for their understanding and patience throughout the writing process. I could not have completed this project without their support.

Contents

Preface

Chapter 1: A Review of Functions

Lesson 1-1: Representing Functions

Lesson 1-2: The Domain of a Function

Lesson 1-3: Operations on Functions

Lesson 1-4: Transformations of Functions

Lesson 1-5: Symmetry

Lesson 1-6: Difference Quotients

Lesson 1-7: Increasing and Decreasing Functions

Chapter 2: Elementary Functions

Lesson 2-1: Linear Functions

Lesson 2-2: Quadratic Functions

Lesson 2-3: Polynomials

Lesson 2-4: Rational Functions

Lesson 2-5: Power Functions

Lesson 2-6: Piecewise-Defined Functions

Chapter 3: Exponential and Logarithmic Functions

Lesson 3-1: Exponential Functions

Lesson 3-2: Inverse Functions

Lesson 3-3: Logarithmic Functions

Lesson 3-4: Exponential and Logarithmic Equations

Chapter 4: Trigonometric Functions

Lesson 4-1: Sine and Cosine

Lesson 4-2: The Tangent Function

Lesson 4-3: Secant, Cosecant and Cotangent Functions

Lesson 4-4: Identities and Formulas

Lesson 4-5: Inverse Trigonometric Functions

Chapter 5: Limits

Lesson 5-1: Evaluating Limits Numerically

Lesson 5-2: Evaluating Limits Graphically

Lesson 5-3: Evaluating Limits Algebraically

Lesson 5-4: Evaluating Limits That Involve Infinity

Chapter 6: Continuity

Lesson 6-1: Continuity and Limits

Lesson 6-2: Types of Discontinuities

Lesson 6-3: The Intermediate Value Theorem

Chapter 7: The Derivative

Lesson 7-1: Secant Lines and Difference Quotients

Lesson 7-2: Tangent Lines

Lesson 7-3: The Definition of the Derivative

Lesson 7-4: The Existence of the Derivative

Lesson 7-5: The Derivative and Tangent Line Equations

Lesson 7-6: Notation

Chapter 8: Rules For Differentiation

Lesson 8-1: Sums and Differences

Lesson 8-2: The Power Rule

Lesson 8-3: The Product Rule

Lesson 8-4: The Quotient Rule

Chapter 9: Derivatives of Exponential and Trigonometric Functions

Lesson 9-1: Exponential Functions

Lesson 9-2: Hyperbolic Functions

Lesson 9-3: Trigonometric Functions

Chapter 10: The Chain Rule

Lesson 10-1: The Power Form of the Chain Rule

Lesson 10-2: The General Chain Rule

Lesson 10-3: Implicit Differentiation

Lesson 10-4: Inverse Functions

Lesson 10-5: Logarithmic Functions

Lesson 10-6: Inverse Trigonometric Functions

Lesson 10-7: Parametric Derivatives

Chapter 11: Applications of the Derivative

Lesson 11-1: Using the Tangent Line

Lesson 11-2: Derivatives and Motion

Lesson 11-3 Differentials

Lesson 11-4 Rolle’s Theorem and the Mean Value Theorem

Chapter 12: Further Applications of the Derivative

Lesson 12-1: Related Rates

Lesson 12-2: L’Hopital’s Rule

Lesson 12-3: The Extreme Value Theorem and Optimization

Chapter 13: Graphical Analysis Using the First Derivative

Lesson 13-1: The First Derivative

Lesson 13-2: Using the First Derivative

Lesson 13-3: Relating the Graphs of f(x) and f′(x)

Chapter 14: Graphical Analysis Using the Second Derivative

Lesson 14-1: The Second Derivative and Concavity

Lesson 14-2: Inflection Points

Lesson 14-3: Using the Second Derivative

Lesson 14-4: Relating the Graphs of f(x), f′(x), and f″(x)

Chapter 15: Integration

Lesson 15-1: Anti-Derivatives

Lesson 15-2: Substitution

Lesson 15-3: Integration by Parts

Lesson 15-4: Integrating Rational Functions

Lesson 15-5: A Strategy for Integration

Lesson 15-6: Area and the Riemann Sum

Lesson 15-7: The Fundamental Theorem of Calculus

Index

About the Author

Preface

Welcome to Homework Helpers: Calculus!

Calculus is a tool that can be used to analyze functions. Two mathematicians, Sir Isaac Newton and Gottfried Leibniz, are credited with developing calculus in the 17th century. There is some controversy regarding the development of calculus. Leibniz published his results before Newton did, and Newton claimed to have developed calculus years earlier but delayed publication. Newton took things a step further and claimed that Leibniz’s work was based on ideas taken from notes and letters written by Newton. It’s hard to imagine anyone fighting over a mathematical result, but politics permeates even the field of mathematics.

Calculus involves the study of limits. The Greeks used limits, or the method of exhaustion, to approximate pi and to compute the area of a region and the volume of a solid. The application of calculus to computing areas and volumes is called integral calculus.

Calculus is also the study of change. Newton’s motivation for developing calculus was to study motion and rates of change as they apply in physics. Calculus can be used to describe Newton’s laws of motion, and Newton’s notation is widely used in physics. The application of calculus to motion and rates of change is called differential calculus.

Calculus marks the transition from working with the static nature of a function to analyzing the dynamic nature of a function. We will be moving away from calculating the value of a function at a particular point and moving towards developing an understanding of how a function changes over a particular interval, or over time.

The ideas in calculus can be explained using everyday language. Calculus problems usually involve one or two steps that actually fall under the realm of calculus. Most of the difficulty in solving calculus problems is algebraic in nature. Calculus problems involve a lot of factoring, multiplying, and dividing polynomials. Solving calculus problems usually requires solving exponential, logarithmic, and trigonometric equations. Calculus will build on the algebraic skills used to analyze a function, not replace them. Because of this, I will spend the first few chapters reviewing the properties of functions from an algebraic perspective. Laying a strong algebraic foundation will definitely pay off in our calculus explorations.

The skills that you will develop in the process of learning calculus can be applied to almost every other field of study imaginable. Physics, chemistry, biology, business, economics, sociology, and medicine are just a few areas where your knowledge of calculus and advanced mathematical problem-solving skills will enable you to excel.

One of the most interesting ideas in calculus has to do with existence theorems. An existence theorem is a claim that an object exists without actually producing the object. The main existence theorems in calculus are the Intermediate Value Theorem, the Extreme Value Theorem, Rolle’s Theorem, and the Mean Value Theorem. Though these theorems are very important results on their own, they will also provide you with some insight into what mathematicians do: We hypothesize, conjecture, and prove theorems. The other important theorem we will discuss is the Fundamental Theorem of Calculus. This theorem bridges the gap between differential calculus and integral calculus.

I wrote this book with the hope that it will help anyone who is struggling to understand calculus or is just curious about the subject. Reading a math book can be a challenge, but I tried to use everyday language to explain the concepts being discussed. Looking at solutions to math problems can sometimes be confusing, so I tried to explain each of the steps I used to get from Point A to Point B. Keep in mind that learning calculus is not a spectator sport. In this book I have worked out many examples, and I have supplied practice problems at the end of most of the lessons. Work these problems out on your own as they come up, and check your answers against the solutions at the end of each chapter. Aside from any typographical errors on my part, our answers should match.

I hope that in reading this book you will develop an appreciation for the subject of calculus and the field of mathematics!

1

A Review of Functions

The concept of a function is very important in mathematics. Functions can be used to describe, or model, many situations in our everyday lives. In economics, functions can be used to calculate income tax, interest earned from an investment, and monthly loan payments. In science, functions can be used to predict the pressure exerted by a gas, the energy released in a chemical reaction, and the occurrence of the next lunar eclipse. The study of calculus requires a solid understanding of some basic elementary functions. It is common for problems in calculus to involve only one or two steps that actually pertain directly to calculus! The majority of the work in solving problems in calculus is algebraic in nature and involves analyzing functions. This chapter should not only serve as a review of the general properties of functions, but also help you understand why it is very important to have a strong foundation in algebra.

Lesson 1-1: Representing Functions

A function is a set of instructions that establishes a relationship between two quantities. A function has input and output values. The input is called the domain and the output is called the range. The variable used to describe the elements in the domain is called the independent variable. The variable used to describe the output is called the dependent variable, as it depends on the input. An important feature of a function is that every input value has only one corresponding output value. A function can be represented in a variety of ways. Functions can be described using words, a formula, a table, or a graph.

Using a formula to define a function is a convenient way to describe the function in mathematical terms instead of using words. When analyzing a formula, it is important to use the order of operations. A function is often written y = f(x), where f(x) is an algebraic expression that involves the independent variable x. The variable x is sometimes referred to as the argument of the function. Evaluating the function for a particular value of x involves replacing every instance of the independent variable with that value.

Scientists often collect data from various instruments and record this information in a table. These tables represent functions, and they provide an easy way to describe a complex, or unknown, formula.

The graph of a function is usually presented using the Cartesian coordinate system. In the Cartesian coordinate system, we use a vertical line, called the y-axis, and a horizontal line, called the x-axis, to divide the plane into four regions, or quadrants. The intersection of these two lines is called the origin.

Two numbers are used to describe the location of a point in the plane, and they are recorded in the form of an ordered pair (x, y). A function can then be thought of as a collection of ordered pairs (x, y). The graph of a function is then the graph of these ordered pairs on the coordinate plane.

If the graph of the function f(x) crosses the x-axis, then we say that f(x) has an x-intercept. The x-intercept of f(x) is the point where f(x) crosses the x-axis. Because any point on the x-axis has a y-coordinate of 0, the x-intercept of f(x) corresponds to a point of the form (a, 0). The x-intercepts of a function are often called the roots of the function, or the zeros of the function. Not all functions have x-intercepts. Finding the x-intercepts of a function involves solving the equation f(x) = 0 for x.

If x = 0 is in the domain of a function f(x), then the point (0, f(0)) is the y-intercept of f(x). The y-intercept of a function represents where the function crosses the y-axis. Not all functions have y-intercepts, but if a function has a y-intercept, the y-intercept will be unique. In other words, a function can have at most one y-intercept.

Lesson 1-2: The Domain of a Function

The domain of a function represents the allowed values of the independent variable. If a function is described using words, then the domain needs to incorporate the context of the description of the function. For example, if a function describes the number of buses needed for a field trip as a function of the number of expected passengers, then the domain of this function cannot include any negative numbers: Transporting a negative number of passengers makes no sense!

The description of a function using a formula may or may not include a domain. If the domain is not indicated, then it is safe to assume that the domain is the set of all real numbers that, when substituted in for the independent variable, produce real values for the dependent variable. To find the domain of a function, start with the set of all real numbers and whittle down the list. There are two things that are frowned on in the mathematical community. The first thing that is forbidden is to divide a non-zero number by 0; quotients such as are meaningless. The second thing that is not allowed in the world of real numbers is to take an even root of a negative number; for example, there is no real number that corresponds to . To find the domain of a function that involves an even root, such as a square root, a fourth root, and so on, set whatever is under the root to be greater than or equal to 0 and find the solutions to the inequality. Then toss out any points that result in the denominator being equal to 0.

Example 1

Find the domain of the function .

Solution: Start with the radical and then deal with the denominator. Set the contents of the radical to be greater than or equal to 0 and solve the inequality:

2 − x ≥0

x ≤ 2

Now focus on the denominator of the function: x + 5 ≠ 0, which means that x ≠ −5. The domain of the function is the set of all real numbers less than or equal to 2, excluding -5.

We can write the domain of a function using interval notation. The table on page 14 summarizes the different types of intervals that we will encounter. Keep in mind that parentheses are always used next to the symbol for infinity, ∞. Brackets are never used next to ∞ because x can never actually reach infinity.

The domain of the function in Example 1, , was the set of all real numbers less than or equal to 2, excluding −5. Using interval notation to describe this subset of the real numbers, we can write the domain as (−∞, −5) (−5, 2].

Lesson 1-2 Review

1. Write the domain of in interval notation.

Lesson 1-3: Operations on Functions

The algebra of real numbers establishes rules for how to combine real numbers. Numbers can be added, subtracted, multiplied, and divided. Algebraic expressions are an abstract way to represent numbers, so it is only natural that we are able to add, subtract, multiply, and divide algebraic expressions as well. There is one additional thing that we can do with functions: We can take their composition.

Think of a function as a transformation of things from the domain, or the input, to things in the range, or the output. If a function f has a domain X and its range is a subset of a set Y, then we use the notation f: X → Y to represent the idea that f is a function from X to Y. Suppose that g is a function whose domain is Y and whose range is a subset of a set Z. We can use the functions f and g to define a new function whose domain is X and whose range is contained in Z. This new function would take an element in X to its corresponding element in Y using the function f and then take that element in Y to an element in Z using the function g. This process of stringing functions from set to set is called the composition of functions. Because f takes things from X to Y, and g takes things from Y to Z, then the new function "g composed with f" takes things directly from X to Z.

We write the composition of f and g in the order described above as g f. The functions are applied right to left: g f(x) means first apply the function f to x, and then apply the function g to the result. We can write g f(x) = g(f(x)); g f(x) is read "g composed with f, or as g of f(x)." The order in which we compose things matters. In general, g f(x)≠ f g(x). In other words, g(f(x)) ≠ f(g(x)).

We can look at a complicated function such as in terms of the composition of two functions. If we define f(x) = 3x + 1 and , then h = g f. Functions are just instructions for what to do with the argument, or the object in parentheses. The function instructs us to take the argument of the function and put it under a radical. The function f(x) = 3x + 1 instructs us to triple the argument and then add 1. So:

Alternatively, we could substitute in for f(x) using its formula and then apply g:

Either way you evaluate g f(x), you get the function h(x).

Now let’s look at the same composition in reverse order. Notice that, in this situation:

Alternatively, if we first substitute in for g(x) using its formula and then apply f, we have:

This illustrates the fact that the order in which you compose functions matters.

In the example we just used, , while .

In general, f g(x) is a different function than g f(x).

Lesson 1-3 Review

Find g f(x) and f g(x) for the following pairs of functions:

Lesson 1-4: Transformations of Functions

The graph of a function can be moved around the coordinate plane. The process by which a graph is moved is referred to as a transformation of a function. In general, the transformation of the graph of a function can involve a shift (also called a translation), a reflection, a stretch, or