Understanding Calculus Problems, Solutions, and Tips

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High School
Topic
Mathematics
Subtopic
Course Workbook
Understanding Calculus:
Problems, Solutions, and Tips
Professor Bruce H. Edwards
University of Florida
Professor 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. Professor Edwards’s coauthored mathematics textbooks have
earned awards from the Text and Academic Authors Association.
Understanding Calculus
Workbook

PUBLISHED BY:
THE GREAT COURSES
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©2010 The Teaching Company. i
Bruce H. Edwards, Ph.D.
Professor of Mathematics, University of Florida
Bruce H. Edwards has been a Professor of Mathematics at the University of Florida since 1976. He received his B.S. in
Mathematics from Stanford University in 1968 and his Ph.D. 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 Bogotá, at
La Universidad Pedagógica y Tecnológica 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. 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. 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.
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 ....................................................................... 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 ............................................. 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 Area.............................................. 101
Lesson Thirty-Three Basic Integration Rules ............................................................................... 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. iii
Table 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