Table Of ContentACTIVE CALCULUS
2016 Edition
Matthew Boelkins
David Austin Steven Schlicker
2
Active Calculus
2016edition
Matthew Boelkins, Lead Author and Editor
Department of Mathematics
Grand Valley State University
[email protected]
http://faculty.gvsu.edu/boelkinm/
David Austin, Contributing Author
http://merganser.math.gvsu.edu/david/
Steven Schlicker, Contributing Author
http://faculty.gvsu.edu/schlicks/
July 19, 2016
ii
©2012-2016Boelkins
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Contents
Preface vii
1 Understanding the Derivative 1
1.1 How do we measure velocity? . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The notion of limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 The derivative of a function at a point . . . . . . . . . . . . . . . . . . . . 22
1.4 The derivative function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5 Interpreting, estimating, and using the derivative . . . . . . . . . . . . . . 43
1.6 The second derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.7 Limits, Continuity, and Differentiability . . . . . . . . . . . . . . . . . . . . 65
1.8 The Tangent Line Approximation . . . . . . . . . . . . . . . . . . . . . . . 77
2 Computing Derivatives 87
2.1 Elementary derivative rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2 The sine and cosine functions . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.3 The product and quotient rules . . . . . . . . . . . . . . . . . . . . . . . . 102
2.4 Derivatives of other trigonometric functions . . . . . . . . . . . . . . . . . 113
2.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.6 Derivatives of Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . 129
2.7 Derivatives of Functions Given Implicitly . . . . . . . . . . . . . . . . . . . 140
2.8 Using Derivatives to Evaluate Limits . . . . . . . . . . . . . . . . . . . . . . 149
3 Using Derivatives 161
3.1 Using derivatives to identify extreme values . . . . . . . . . . . . . . . . . 161
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iv CONTENTS
3.2 Using derivatives to describe families of functions . . . . . . . . . . . . . . 174
3.3 Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.4 Applied Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.5 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4 The Definite Integral 207
4.1 Determining distance traveled from velocity . . . . . . . . . . . . . . . . . 207
4.2 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.4 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . 250
5 Finding Antiderivatives and Evaluating Integrals 265
5.1 Constructing Accurate Graphs of Antiderivatives . . . . . . . . . . . . . . 265
5.2 The Second Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 276
5.3 Integration by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
5.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
5.5 Other Options for Finding Algebraic Antiderivatives . . . . . . . . . . . . 310
5.6 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6 Using Definite Integrals 333
6.1 Using Definite Integrals to Find Area and Length . . . . . . . . . . . . . . 333
6.2 Using Definite Integrals to Find Volume . . . . . . . . . . . . . . . . . . . . 343
6.3 Density, Mass, and Center of Mass . . . . . . . . . . . . . . . . . . . . . . . 354
6.4 Physics Applications: Work, Force, and Pressure . . . . . . . . . . . . . . . 365
6.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7 Differential Equations 387
7.1 An Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . 387
7.2 Qualitative behavior of solutions to DEs . . . . . . . . . . . . . . . . . . . 398
7.3 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
7.4 Separable differential equations . . . . . . . . . . . . . . . . . . . . . . . . 419
7.5 Modeling with differential equations . . . . . . . . . . . . . . . . . . . . . . 427
7.6 Population Growth and the Logistic Equation . . . . . . . . . . . . . . . . 435
CONTENTS v
8 Sequences and Series 447
8.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
8.2 Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
8.3 Series of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
8.4 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
8.5 Taylor Polynomials and Taylor Series . . . . . . . . . . . . . . . . . . . . . 501
8.6 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518
A A Short Table of Integrals 531
vi CONTENTS
Preface
A free and open-source calculus
Several fundamental ideas in calculus are more than 2000 years old. As a formal
subdisciplineofmathematics, calculuswasfirstintroducedanddevelopedinthelate1600s,
with key independent contributions from Sir Isaac Newton and Gottfried Wilhelm Leibniz.
Mathematicians agree that the subject has been understood rigorously since the work of
Augustin Louis Cauchy and Karl Weierstrass in the mid 1800s when the field of modern
analysis was developed, in part to make sense of the infinitely small quantities on which
calculus rests. Hence, as a body of knowledge calculus has been completely understood
by experts for at least 150 years. The discipline is one of our great human intellectual
achievements: among many spectacular ideas, calculus models how objects fall under
the forces of gravity and wind resistance, explains how to compute areas and volumes of
interesting shapes, enables us to work rigorously with infinitely small and infinitely large
quantities, and connects the varying rates at which quantities change to the total change
in the quantities themselves.
While each author of a calculus textbook certainly offers her own creative perspective
onthesubject,itishardlythecasethatmanyoftheideasshepresentsarenew. Indeed,the
mathematics community broadly agrees on what the main ideas of calculus are, as well as
their justification and their importance; the core parts of nearly all calculus textbooks are
very similar. As such, it is our opinion that in the 21st century – an age where the internet
permits seamless and immediate transmission of information – no one should be required
to purchase a calculus text to read, to use for a class, or to find a coherent collection of
problemstosolve. Calculusbelongstohumankind, notanyindividualauthororpublishing
company. Thus, a main purpose of this work is to present a new calculus text that is free.
In addition, instructors who are looking for a calculus text should have the opportunity to
download the source files and make modifications that they see fit; thus this text is open-
source. Since August 2013, Active Calculus has been endorsed by the American Institute of
Mathematics and its Open Textbook Initiative: http://aimath.org/textbooks/.
Because the text is free, any professor or student may use the electronic version of the
text for no charge. A .pdf copy of the text may be obtained by download from
http://gvsu.edu/s/xr,
vii
viii
where the user will also find a link to a print-on-demand for purchasing a bound, softcover
version for about $20. Other ancillary materials, such as WeBWorK .def files, an activities-
onlyworkbook,andsamplecomputerlaboratoryactivitiesareavailableupondirectrequest
to the author. Furthermore, because the text is open-source, any instructor may acquire
the full set of source files, again by request to the author at [email protected]. This
work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0
Unported License. The graphic
that appears throughout the text shows that the work is licensed with the Creative
Commons, that the work may be used for free by any party so long as attribution is given
to the author(s), that the work and its derivatives are used in the spirit of “share and share
alike,” and that no party may sell this work or any of its derivatives for profit, with the
following exception: it is entirely acceptable for university bookstores to sell bound photocopied
copies of the activities workbook to students at their standard markup above the copying expense.
Full details may be found by visiting
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or sending a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View,
California, 94041, USA.
Active Calculus: our goals
In Active Calculus, we endeavor to actively engage students in learning the subject through
an activity-driven approach in which the vast majority of the examples are completed by
students. Where many texts present a general theory of calculus followed by substantial
collections of worked examples, we instead pose problems or situations, consider possibili-
ties, and then ask students to investigate and explore. Following key activities or examples,
thepresentationnormallyincludessomeoverallperspectiveandabriefsynopsisofgeneral
trends or properties, followed by formal statements of rules or theorems. While we often
offer plausibility arguments for such results, rarely do we include formal proofs. It is not
the intent of this text for the instructor or author to demonstrate to students that the ideas
of calculus are coherent and true, but rather for students to encounter these ideas in a
supportive, leading manner that enables them to begin to understand for themselves why
calculus is both coherent and true. This approach is consistent with the growing body of
scholarship that calls for students to be interactively engaged in class.
