Calculus.pdf
The right way to begin a calculus book is with calculus. This chapter will jump (That is integration and it is the goal of integral calculus.).
Calculus Volume 1 - OpenStax
- If you use this textbook as a bibliographic reference please include https://openstax.org/details/books/calculus-volume-1 in your citation. For questions
Calculus Volume 1
- If you use this textbook as a bibliographic reference please include https://openstax.org/details/books/calculus-volume-. 1 in your citation. For questions
ADVANCED CALCULUS
Spivak and Pure Mathematics by. G. Hardy. The reader should also have some experience with partial derivatives. In overall plan the book divides roughly into a
Single and Multivariable Calculus
If you distribute this work or a derivative include the history of the document. This text was initially written by David Guichard. The single variable
Single and Multivariable Calculus
If you distribute this work or a derivative include the history of the document. This text was initially written by David Guichard. The single variable
The AP Calculus Problem Book ?
The AP Calculus Problem Book. Publication history: First edition 2002. Second edition
Shana Calaway Dale Hoffman David Lippman
He is the coauthor of the open textbooks Precalculus: An Investigation of Functions and Math in Society. Page 3. Introduction. Business Calculus. 3.
OSU Briggs Calculus Book Buying Guide – 2017-2018 School Year
OSU Calculus courses not listed above use other textbooks. 4. eBook means electronic online textbook inside MyMathLab. 5. If you are repeating a course from
Refinement Calculus: - LARA
The second part of the book describes the predicate transformer approach to programming logic and program semantics as well as the notion of program refinement
Single and Multivariable
Calculus
Early Transcendentals
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit or send a letter toCreative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. If you distribute
this work or a derivative, include the history of the document.This text was initially written by David Guichard. The single variable material in chapters 1-9 is a mod-
ification and expansion of notes written by Neal Koblitz at the University of Washington, who generously
gave permission to use, modify, and distribute his work. New material has been added, and old material
has been modified, so some portions now bear little resemblance to the original. The book includes some exercises and examples fromElementary Calculus: An Approach Using Infinitesi- mals, by H. Jerome Keisler, available at under a CreativeCommons license. In addition, the chapter on differential equations (in the multivariable version) and the
section on numerical integration are largely derived from the corresponding portions of Keisler's book.
Some exercises are from the OpenStax Calculus books, available free at https://openstax.org/subjects/math Albert Schueller, Barry Balof, and Mike Wills have contributed additional material. This copy of the text was compiled from source at 14:34 on 12/1/2022.The current version of the text is available at
I will be glad to receive corrections and suggestions for improvement atguichard@whitman.edu.For Kathleen,
without whose encouragement this book would not have been written.Contents
1Analytic Geometry
151.1Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
161.2Distance Between Two Points; Circles . . . . . . . . . . . . . . .
211.3Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
221.4Shifts and Dilations . . . . . . . . . . . . . . . . . . . . . . . .
272
Instantaneous Rate of Change: The Derivative
312.1The slope of a function . . . . . . . . . . . . . . . . . . . . . .
312.2An example . . . . . . . . . . . . . . . . . . . . . . . . . . . .
362.3Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
382.4The Derivative Function . . . . . . . . . . . . . . . . . . . . .
482.5Properties of Functions . . . . . . . . . . . . . . . . . . . . . .
535
6Contents
3Rules for Finding Derivatives
573.1The Power Rule . . . . . . . . . . . . . . . . . . . . . . . . .
573.2Linearity of the Derivative . . . . . . . . . . . . . . . . . . . .
603.3The Product Rule . . . . . . . . . . . . . . . . . . . . . . . .
623.4The Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . .
643.5The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . .
674
Transcendental Functions
734.1Trigonometric Functions . . . . . . . . . . . . . . . . . . . . .
734.2The Derivative of sinx. . . . . . . . . . . . . . . . . . . . . .
764.3A hard limit . . . . . . . . . . . . . . . . . . . . . . . . . . .
774.4The Derivative of sinx, continued . . . . . . . . . . . . . . . . .
804.5Derivatives of the Trigonometric Functions . . . . . . . . . . . .
814.6Exponential and Logarithmic functions . . . . . . . . . . . . . .
824.7Derivatives of the exponential and logarithmic functions . . . . .
844.8Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . .
894.9Inverse Trigonometric Functions . . . . . . . . . . . . . . . . .
944.10Limits revisited . . . . . . . . . . . . . . . . . . . . . . . . . .
974.11 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 102
5
Curve Sketching
1075.1Maxima and Minima . . . . . . . . . . . . . . . . . . . . . .
1075.2The first derivative test . . . . . . . . . . . . . . . . . . . . .
1115.3The second derivative test . . . . . . . . . . . . . . . . . . .
1135.4Concavity and inflection points . . . . . . . . . . . . . . . . .
1145.5Asymptotes and Other Things to Look For . . . . . . . . . . .
116Contents7
6Applications of the Derivative
1196.1Optimization . . . . . . . . . . . . . . . . . . . . . . . . . .
1196.2Related Rates . . . . . . . . . . . . . . . . . . . . . . . . .
1316.3Newton's Method . . . . . . . . . . . . . . . . . . . . . . . .
1396.4Linear Approximations . . . . . . . . . . . . . . . . . . . . .
1436.5The Mean Value Theorem . . . . . . . . . . . . . . . . . . .
1457
Integration
1497.1Two examples . . . . . . . . . . . . . . . . . . . . . . . . .
1497.2The Fundamental Theorem of Calculus . . . . . . . . . . . . .
1537.3Some Properties of Integrals . . . . . . . . . . . . . . . . . .
1608
Techniques of Integration
1658.1Substitution . . . . . . . . . . . . . . . . . . . . . . . . . .
1668.2Powers of sine and cosine . . . . . . . . . . . . . . . . . . . .
1718.3Trigonometric Substitutions . . . . . . . . . . . . . . . . . . .
1738.4Integration by Parts . . . . . . . . . . . . . . . . . . . . . .
1768.5Rational Functions . . . . . . . . . . . . . . . . . . . . . . .
1808.6Numerical Integration . . . . . . . . . . . . . . . . . . . . . .
1848.7Additional exercises . . . . . . . . . . . . . . . . . . . . . . .
1898Contents
9Applications of Integration
1919.1Area between curves . . . . . . . . . . . . . . . . . . . . . .
1919.2Distance, Velocity, Acceleration . . . . . . . . . . . . . . . . .
1969.3Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1999.4Average value of a function . . . . . . . . . . . . . . . . . . .
2069.5Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2099.6Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . .
2139.7Kinetic energy; improper integrals . . . . . . . . . . . . . . .
2189.8Probability . . . . . . . . . . . . . . . . . . . . . . . . . . .
2229.9Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . .
2329.10Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . .
23410
Polar Coordinates, Parametric Equations
23910.1Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . .
23910.2Slopes in polar coordinates . . . . . . . . . . . . . . . . . . .
24310.3Areas in polar coordinates . . . . . . . . . . . . . . . . . . .
24510.4Parametric Equations . . . . . . . . . . . . . . . . . . . . . .
24810.5Calculus with Parametric Equations . . . . . . . . . . . . . .
251Contents9
11Sequences and Series
25511.1Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25611.2Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26211.3The Integral Test . . . . . . . . . . . . . . . . . . . . . . . .
26611.4Alternating Series . . . . . . . . . . . . . . . . . . . . . . . .
27111.5Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . .
27311.6Absolute Convergence . . . . . . . . . . . . . . . . . . . . .
27611.7The Ratio and Root Tests . . . . . . . . . . . . . . . . . . .
27711.8Power Series . . . . . . . . . . . . . . . . . . . . . . . . . .
28011.9Calculus with Power Series . . . . . . . . . . . . . . . . . . .
28311.10Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . .
28511.11Taylor's Theorem . . . . . . . . . . . . . . . . . . . . . . . .
28811.12Additional exercises . . . . . . . . . . . . . . . . . . . . . . .
29412
Three Dimensions
29712.1The Coordinate System . . . . . . . . . . . . . . . . . . . . .
29712.2Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30012.3The Dot Product . . . . . . . . . . . . . . . . . . . . . . . .
30612.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . 312
12.5Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . .
31612.6Other Coordinate Systems . . . . . . . . . . . . . . . . . . .
32313
Vector Functions
32913.1Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . .
32913.2Calculus with vector functions . . . . . . . . . . . . . . . . .
33113.3Arc length and curvature . . . . . . . . . . . . . . . . . . . .
33913.4Motion along a curve . . . . . . . . . . . . . . . . . . . . . .
34510Contents
14Partial Differentiation
34914.1Functions of Several Variables . . . . . . . . . . . . . . . . .
34914.2Limits and Continuity . . . . . . . . . . . . . . . . . . . . .
35314.3Partial Differentiation . . . . . . . . . . . . . . . . . . . . . .
35714.4The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . .
36414.5Directional Derivatives . . . . . . . . . . . . . . . . . . . . .
36714.6Higher order derivatives . . . . . . . . . . . . . . . . . . . . .
37214.7Maxima and minima . . . . . . . . . . . . . . . . . . . . . .
37314.8Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . .
37915
Multiple Integration
38515.1Volume and Average Height . . . . . . . . . . . . . . . . . .
38515.2 Double Integrals in Cylindrical Coordinates . . . . . . . . . . . 395
15.3Moment and Center of Mass . . . . . . . . . . . . . . . . . .
40015.4Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . .
40215.5Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . .
40415.6Cylindrical and Spherical Coordinates . . . . . . . . . . . . .
40715.7Change of Variables . . . . . . . . . . . . . . . . . . . . . . .
41116
Vector Calculus
41916.1Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . .
41916.2Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . .
42116.3The Fundamental Theorem of Line Integrals . . . . . . . . . .
42516.4Green's Theorem . . . . . . . . . . . . . . . . . . . . . . . .
42816.5Divergence and Curl . . . . . . . . . . . . . . . . . . . . . .
43316.6Vector Functions for Surfaces . . . . . . . . . . . . . . . . . .
43616.7Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . .
44216.8Stokes's Theorem . . . . . . . . . . . . . . . . . . . . . . . .
44616.9The Divergence Theorem . . . . . . . . . . . . . . . . . . . .
450Contents11
17Differential Equations
45517.1First Order Differential Equations . . . . . . . . . . . . . . .
45617.2First Order Homogeneous Linear Equations . . . . . . . . . . .
46017.3First Order Linear Equations . . . . . . . . . . . . . . . . . .
46317.4Approximation . . . . . . . . . . . . . . . . . . . . . . . . .
46617.5Second Order Homogeneous Equations . . . . . . . . . . . . .
46917.6Second Order Linear Equations . . . . . . . . . . . . . . . . .
47317.7Second Order Linear Equations, take two . . . . . . . . . . . .
477A
Selected Answers
481B
Useful Formulas
509Index 513
Introduction
The emphasis in this course is on problems - doing calculations and story problems. To master problem solving one needs a tremendous amount of practice doing problems. The more problems you do the better you will be at doing them, as patterns will start to emerge in both the problems and in successful approaches to them. You will learn fastest and best if you devote some time to doing problems every day. Typically the most difficult problems are story problems, since they require some effort before you can begin calculating. Here are some pointers for doing story problems:1.Carefully read each problem twice before writing anything.
2.Assign letters to quantities that are described only in words; draw a diagram if
appropriate.3.Decide which letters are constants and which are variables. A letter stands for a
constant if its value remains the same throughout the problem.4.Using mathematical notation, write down what you know and then write down
what you want to find.5.Decide what category of problem it is (this might be obvious if the problem comes
at the end of a particular chapter, but will not necessarily be so obvious if it comes on an exam covering several chapters).6.Double check each step as you go along; don't wait until the end to check your
work. 7. Use common sense; if an answer is out of the range of practical possibilities, then check your work to see where you went wrong. 1314Introduction
Suggestions for Using This Text
1.Read the example problems carefully, filling in any steps that are left out (ask
someone for help if you can't follow the solution to a worked example).2.Later use the worked examples to study by covering the solutions, and seeing if
you can solve the problems on your own.3.Most exercises have answers in Appendix
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