### Transcription of Early Transcendentals an Open Text - Lyryx Learning

1 With **open** **texts** CALCULUS. **Early** **Transcendentals** an **open** Text BASE TEXTBOOK. VERSION 2017 REVISION A. ADAPTABLE | ACCESSIBLE | AFFORDABLE. by **Lyryx** **Learning** based on the original text by D. Guichard Creative Commons License (CC BY-NC-SA). a d v a n c i n g l e a r n i n g Champions of Access to Knowledge **open** TEXT ONLINE. ASSESSMENT. All digital forms of access to our high-quality We have been developing superior online for- **open** **texts** are entirely FREE! All content is mative assessment for more than 15 years. Our reviewed for excellence and is wholly adapt- questions are continuously adapted with the able; custom editions are produced by **Lyryx** content and reviewed for quality and sound for those adopting **Lyryx** assessment. Access pedagogy. To enhance **Learning** , students re- to the original source files is also **open** to any- ceive immediate personalized feedback. Stu- one! dent grade reports and performance statistics are also provided.

2 SUPPORT INSTRUCTOR. SUPPLEMENTS. Access to our in-house support team is avail- Additional instructor resources are also freely able 7 days/week to provide prompt resolution accessible. Product dependent, these supple- to both student and instructor inquiries. In ad- ments include: full sets of adaptable slides and dition, we work one-on-one with instructors to lecture notes, solutions manuals, and multiple provide a comprehensive system, customized choice question banks with an exam building for their course. This can include adapting the tool. text, managing multiple sections, and more! Contact **Lyryx** Today! a d v a n c i n g l e a r n i n g Calculus **Early** **Transcendentals** an **open** Text BE A CHAMPION OF OER! Contribute suggestions for improvements, new content, or errata: A new topic A new example An interesting new question A new or better proof to an existing theorem Any other suggestions to improve the material Contact **Lyryx** at with your ideas.

3 CONTRIBUTIONS. Jim Bailey, College of the Rockies Mark Blenkinsop, Carleton University Michael Cavers, University of Calgary David Guichard, Whitman College APEX Calculus: Gregory Hartman, Virginia Military Institute Joseph Ling, University of Calgary **Lyryx** **Learning** Team Bruce Bauslaugh Jennifer MacKenzie Peter Chow Tamsyn Murnaghan Nathan Friess Bogdan Sava Stephanie Keyowski Larissa Stone Claude Laflamme Ryan Yee Martha Laflamme Ehsun Zahedi LICENSE. Creative Commons License (CC BY-NC-SA): This text, including the art and illustrations, are available under the Creative Commons license (CC BY-NC-SA), allowing anyone to reuse, revise, remix and redistribute the text. To view a copy of this license, visit a d v a n c i n g l e a r n i n g Calculus **Early** **Transcendentals** an **open** Text Base Text Revision History Current Revision: Version 2017 Revision A. Extensive edits, additions, and revisions have been completed by the editorial team at **Lyryx** **Learning** .

4 All new content (text and images) is released under the same license as noted above. **Lyryx** : Front matter has been updated including cover page, copyright, and revision pages. **Lyryx** : Examples , , , , , and Exercises , , have been 2017 A. rewritten. Several exercises from Vector Calculus have been removed. **Lyryx** : Order and name of topics in Chapter 15 and Chapter 16 have been revised. D. Guichard: New content developed for the Three Dimensions, Vector Functions, and Vector Calcu- lus chapters. **Lyryx** : Exercise numbering has been updated to restart with each section. G. Hartman: New content on Riemann Sums is included, Section This section was adapted 2016 B. by **Lyryx** from the section of the same name in APEX Calculus, Version , written by G. Hart- man. T. Siemers and D. Chalishajar of the Virginia Military Instititue and B. Heinold of Mount Saint Mary's University also contributed to APEX Calculus.

5 This material is released under Creative Commons license CC BY-NC ( ). See for more information and original version. **Lyryx** : The layout and appearance of the text has been updated, including the title page and newly 2016 A. designed back cover. J. Ling: Addition of new exercises and proofs throughout. J. Ling: Revised arrangement of topics in the Application of Derivatives chapter. 2015 A. J. Ling: Continuity section has been revised to include additional explanations of content and addi- tional examples. M. Cavers: Addition of new material and images particularly in the Review chapter. 2014 A. M. Blenkinsop: Addition of content including Linear and Higher Order Approximations section. Original text by D. Guichard of Whitman College, the single variable material is a modifi- cation and expansion of notes written and released by N. Koblitz of the University of Wash- ington. That version also contains exercises and examples from Elementary Calculus: An Approach Using Infinitesimals, written by H.

6 J. Keisler of the University of Wisconsin un- 2012 A der a Creative Commons license (see ~ ). A. Schueller, B. Balof, and M. Wills all of Whitman College, have also contributed con- tent. This material is released under the Creative Commons Attribution-NonCommercial- ShareAlike License ( ). See for more information. Contents Contents iii Introduction 1. 1 Review 3. Algebra .. 3. Sets and Number Systems .. 3. Law of Exponents .. 5. The Quadratic Formula and Completing the Square .. 6. Inequalities, Intervals and Solving Basic Inequalities .. 8. The Absolute Value .. 14. Solving Inequalities that Contain Absolute Values .. 15. Analytic Geometry .. 18. Lines .. 19. Distance between Two Points and Midpoints .. 24. Conics .. 25. Trigonometry .. 32. Angles and Sectors of Circles .. 32. Trigonometric Functions .. 33. Computing Exact Trigonometric Ratios .. 35. Graphs of Trigonometric Functions .. 40.

7 Trigonometric Identities .. 40. Additional Exercises .. 43. 2 Functions 45. What is a Function? .. 45. Transformations and Compositions .. 50. Transformations .. 50. Combining Two Functions .. 52. Exponential Functions .. 54. Inverse Functions .. 57. Logarithms .. 60. Inverse Trigonometric Functions .. 63. iii iv Contents Hyperbolic Functions .. 68. Additional Exercises .. 72. 3 Limits 75. The Limit .. 75. Precise Definition of a Limit .. 77. Computing Limits: Graphically .. 80. Computing Limits: Algebraically .. 82. Infinite Limits and Limits at Infinity .. 86. Vertical Asymptotes .. 89. Horizontal Asymptotes .. 90. Slant Asymptotes .. 91. End Behaviour and Comparative Growth Rates .. 92. A Trigonometric Limit .. 98. Continuity .. 102. 4 Derivatives 115. The Rate of Change of a Function .. 115. The Derivative Function .. 121. Differentiable .. 125. Second and Other Derivatives .. 127. Velocities.

8 128. Derivative Rules .. 130. Derivative Rules for Trigonometric Functions .. 135. The Chain Rule .. 137. Derivatives of Exponential & Logarithmic Functions .. 142. Implicit Differentiation .. 148. Derivatives of Inverse Functions .. 156. Derivatives of Inverse Trigonometric Functions .. 157. Additional Exercises .. 159. 5 Applications of Derivatives 163. Related Rates .. 163. Extrema of a Function .. 169. Local Extrema .. 169. Absolute Extrema .. 174. The Mean Value Theorem .. 178. Linear and Higher Order Approximations .. 184. Linear Approximations .. 184. Contents v Differentials .. 186. Taylor Polynomials .. 188. Newton's Method .. 190. L'H pital's Rule .. 194. Curve Sketching .. 199. Intervals of Increase/Decrease, and the First Derivative Test .. 199. The Second Derivative Test .. 201. Concavity and Inflection Points .. 203. Asymptotes and Other Things to Look For .. 205. Summary of Curve Sketching.

9 206. Optimization Problems .. 210. 6 Integration 219. Displacement and Area .. 219. Riemann Sums .. 222. The Fundamental Theorem of Calculus .. 237. Indefinite Integrals .. 248. 7 Techniques of Integration 253. Substitution Rule .. 253. Powers of Trigonometric Functions .. 260. Trigonometric Substitutions .. 270. Integration by Parts .. 277. Rational Functions .. 281. Numerical Integration .. 286. Improper Integrals .. 291. Additional exercises .. 300. 8 Applications of Integration 303. Distance, Velocity, Acceleration .. 303. Area Between Curves .. 306. Volume .. 312. Average Value of a Function .. 319. Work .. 322. Center of Mass .. 326. Arc Length .. 331. Surface Area .. 334. vi Contents 9 Sequences and Series 339. Sequences .. 340. Series .. 347. The Integral Test .. 351. Alternating Series .. 356. Comparison Tests .. 358. Absolute Convergence .. 361. The Ratio and Root Tests .. 362. Power Series.

10 365. Calculus with Power Series .. 368. Taylor Series .. 370. Taylor's Theorem .. 374. 10 Differential Equations 379. First Order Differential Equations .. 379. First Order Homogeneous Linear Equations .. 384. First Order Linear Equations .. 387. Approximation .. 389. Second Order Homogeneous Equations .. 393. Second Order Linear Equations - Method of Undetermined Coefficients .. 397. Second Order Linear Equations -Variation of Parameters .. 401. 11 Polar Coordinates, Parametric Equations 405. Polar Coordinates .. 405. Slopes in Polar Coordinates .. 409. Areas in Polar Coordinates .. 411. Parametric Equations .. 415. Calculus with Parametric Equations .. 417. Conics in Polar Coordinates .. 419. 12 Three Dimensions 425. The Coordinate System .. 425. Vectors .. 429. The Dot Product .. 433. The Cross Product .. 440. Lines and Planes .. 443. Other Coordinate Systems .. 450. Contents vii 13 Partial Differentiation 457.