Understanding Calculus: Problems, Solutions, and Tips

1 THE GREAT COURSES Corporate Headquarters4840 Westfields Boulevard, Suite 500 Chantilly, VA 20151-2299 USAP hone: No. 1007 2010 The Teaching Image: Image pure intellectual stimulation that can be popped into the [audio or video player] anytime. Harvard Magazine Passionate, erudite, living legend lecturers. Academia s best lecturers are being captured on tape. The Los Angeles Times A serious force in American education. The Wall Street JournalHigh SchoolTopicMathematicsSubtopicCourse WorkbookUnderstanding calculus : Problems, solutions , and TipsProfessor Bruce H. EdwardsUniversity of FloridaProfessor Bruce H. Edwards is Professor of mathematics at the University of Florida, where he has won a host of awards and recognitions. He was named Teacher of the Year in the College of Arts and Sciences and was selected as a Distinguished Alumni Professor by the Office of Alumni Affairs.

2 Professor Edwards s coauthored mathematics textbooks have earned awards from the Text and Academic Authors CalculusWorkbookPUBLISHED BY:THE GREAT COURSESC orporate Headquarters4840 Westfi elds Boulevard, Suite 500 Chantilly, Virginia 20151-2299 Phone: 1-800-832-2412 Fax: The Teaching Company, 2010 Printed in the United States of AmericaThis book is in copyright. All rights reserved. Without limiting the rights under copyright reserved above,no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission ofThe Teaching Company. 2010 The Teaching Company. iBruce H. Edwards, Professor of mathematics , University of Florida Bruce H.

3 Edwards has been a Professor of mathematics at the University of Florida since 1976. He received his in mathematics from Stanford University in 1968 and his in mathematics from Dartmouth College in 1976. From 1968 to 1972, he was a Peace Corps volunteer in Colombia, where he taught mathematics (in Spanish) near Bogot , at La Universidad Pedag gica y Tecnol gica de Colombia. Professor Edwards s early research interests were in the broad area of pure mathematics called algebra. His dissertation in quadratic forms was titled Induction Techniques and Periodicity in Clifford Algebras. Beginning in 1978, he became interested in applied mathematics while working summers for NASA at the Langley Research Center in Virginia. This led to his research in the area of numerical analysis and the solution of differential equations.

4 During his sabbatical year 1984 1985, he worked on 2-point boundary value problems with Professor Leo Xanthis at the Polytechnic of Central London. Professor Edwards s current research is focused on the algorithm called CORDIC that is used in computers and graphing calculators for calculating function values. Professor Edwards has coauthored a wide range of mathematics textbooks with Professor Ron Larson of Penn State Erie, The Behrend College. They have published leading texts in the areas of calculus , applied calculus , linear algebra, finite mathematics , algebra, trigonometry, and precalculus. This course is based on the bestselling textbook calculus (9th edition, Brooks/Cole, 2010). Professor Edwards has won many teaching awards at the University of Florida. He was named Teacher of the Year in the College of Liberal Arts and Sciences in 1979, 1981, and 1990.

5 He was both the Liberal Arts and Sciences Student Council Teacher of the Year and the University of Florida Honors Program Teacher of the Year in 1990. He was also selected by the alumni affairs office to be the Distinguished Alumni Professor for 1991 1993. The winners of this 2-year award are selected by graduates of the university. The Florida Section of the Mathematical Association of America awarded him the Distinguished Service Award in 1995 for his work in mathematics education for the state of Florida. Finally, his textbooks have been honored with various awards from the Text and Academic Authors Association. Professor Edwards has been a frequent speaker at both research conferences and meetings of the National Council of Teachers of mathematics . He has spoken on issues relating to the Advanced Placement calculus examination, especially the use of graphing calculators.

6 Professor Edwards has taught a wide range of mathematics courses at the University of Florida, from first-year calculus to graduate-level classes in algebra and numerical analysis. He particularly enjoys teaching calculus to freshman, due to the beauty of the subject and the enthusiasm of the students. 2010 The Teaching Company. ii Table of Contents Understanding calculus : Problems, solutions , and Tips Professor Biography .. i Course Scope .. 1 Lesson One A Preview of calculus .. 3 Lesson Two Review Graphs, Models, and Functions .. 5 Lesson Three Review Functions and Trigonometry .. 8 Lesson Four Finding Limits .. 11 Lesson Five An Introduction to Continuity.

7 15 Lesson Six Infinite Limits and Limits at Infinity .. 18 Lesson Seven The Derivative and the Tangent Line problem .. 21 Lesson Eight Basic Differentiation Rules .. 24 Lesson Nine Product and Quotient Rules .. 27 Lesson Ten The Chain Rule .. 30 Lesson Eleven Implicit Differentiation and Related Rates .. 32 Lesson Twelve Extrema on an Interval .. 35 Lesson Thirteen Increasing and Decreasing Functions .. 38 Lesson Fourteen Concavity and Points of Inflection .. 42 Lesson Fifteen Curve Sketching and Linear Approximations .. 45 Lesson Sixteen Applications Optimization Problems, Part 1 .. 48 Lesson Seventeen Applications Optimization Problems, Part 2 .. 50 Lesson Eighteen Antiderivatives and Basic Integration Rules .. 53 Lesson Nineteen The Area problem and the Definite Integral .. 56 Lesson Twenty The Fundamental Theorem of calculus , Part 1.

8 61 Lesson Twenty-One The Fundamental Theorem of calculus , Part 2 .. 64 Lesson Twenty-Two Integration by Substitution .. 67 Lesson Twenty-Three Numerical Integration .. 70 Lesson Twenty-Four Natural Logarithmic Function Differentiation .. 73 Lesson Twenty-Five Natural Logarithmic Function Integration .. 76 Lesson Twenty-Six Exponential Function .. 79 Lesson Twenty-Seven Bases other than e .. 82 Lesson Twenty-Eight Inverse Trigonometric Functions .. 86 Lesson Twenty-Nine Area of a Region between 2 Curves .. 90 Lesson Thirty Volume The Disk Method .. 94 Lesson Thirty-One Volume The Shell Method .. 97 Lesson Thirty-Two Applications Arc Length and Surface 101 Lesson Thirty-Three Basic Integration Rules.

9 104 Lesson Thirty-Four Other Techniques of Integration .. 107 Lesson Thirty-Five Differential Equations and Slope Fields .. 110 Lesson Thirty-Six Applications of Differential Equations .. 113 2010 The Teaching Company. iiiTable of Contents Understanding calculus : Problems, solutions , and Tips Glossary .. 115 Formulas .. 121 Theorems .. 124 Review of Trigonometry .. 126 Bibliography .. 128 solutions .. 129 Summary Sheet .. 213 2010 The Teaching Company. iv 2010 The Teaching Company. Understanding calculus : Problems, solutions , and Tips Scope: The goal of this course is for you to understand and appreciate the beautiful subject of calculus . You will see how calculus plays a fundamental role in all of science and engineering, as well as business and economics. You will learn about the 2 major ideas of calculus the derivative and the integral.

10 Each has a rich history and many practical applications. calculus is often described as the mathematics of change. For instance, calculus is the mathematics of velocities, accelerations, tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and a variety of other concepts that have enabled scientists, engineers, and economists to model real-life situations. For example, a NASA scientist might need to know the initial velocity required for a rocket to escape Earth s gravitational field. calculus is required to determine this escape velocity. An engineer might need to know the volume of a spherical object with a hole drilled through the center. The integral calculus is needed to compute this volume. calculus is an important tool for economic predictions, such as the growth of the federal debt.