APEX CALCULUS II Late Transcendentals University of North Dakota Adapted from APEX Calculus by Gregory Hartman, Ph.D. Department of Applied Mathematics Virginia Military Institute Revised June 2021 Contributing Authors Troy Siemers, Ph.D. Michael Corral Department of Applied Mathematics Mathematics Virginia Military Institute Schoolcraft College Brian Heinold, Ph.D. Paul Dawkins, Ph.D. Department of Mathematics Department of Mathematics and Computer Science Lamar University Mount Saint Mary’s University Dimplekumar Chalishajar, Ph.D. Department of Applied Mathematics Virginia Military Institute Editor Jennifer Bowen, Ph.D. Department of Mathematics and Computer Science The College of Wooster Copyright © 2015 Gregory Hartman © 2021 Department of Mathematics, University of North Dakota This work is licensed under a Creative Commons Attribution‐NonCommercial 4.0 International License. Resale and reproduction restricted. CONTENTS Preface vii Calculus I 1 Limits 7 1.1 An Introduction To Limits .................... 8 1.2 Epsilon‐Delta Definition of a Limit ............... 16 1.3 Finding Limits Analytically .................... 25 1.4 One Sided Limits ........................ 39 1.5 Limits Involving Infinity ..................... 48 1.6 Continuity ............................ 60 2 Derivatives 79 2.1 Instantaneous Rates of Change: The Derivative ........ 79 2.2 Interpretations of the Derivative ................ 94 2.3 Basic Differentiation Rules ................... 101 2.4 The Product and Quotient Rules ................ 110 2.5 The Chain Rule ......................... 122 2.6 Implicit Differentiation ..................... 133 3 The Graphical Behavior of Functions 143 3.1 Extreme Values ......................... 143 3.2 The Mean Value Theorem .................... 152 3.3 Increasing and Decreasing Functions .............. 158 3.4 Concavity and the Second Derivative .............. 169 3.5 Curve Sketching ......................... 177 4 Applications of the Derivative 187 4.1 Related Rates .......................... 187 4.2 Optimization ........................... 195 4.3 Differentials ........................... 203 4.4 Newton’s Method ........................ 211 5 Integration 219 5.1 Antiderivatives and Indefinite Integration ........... 219 5.2 The Definite Integral ...................... 229 5.3 Riemann Sums ......................... 240 5.4 The Fundamental Theorem of Calculus ............. 261 5.5 Substitution ........................... 276 6 Applications of Integration 291 6.1 Area Between Curves ...................... 292 6.2 Volume by Cross‐Sectional Area; Disk and Washer Methods .. 300 6.3 The Shell Method ........................ 312 6.4 Work ............................... 322 6.5 Fluid Forces ........................... 332 Calculus II 7 Inverse Functions and L’Hôpital’s Rule 343 7.1 Inverse Functions ........................ 343 7.2 Derivatives of Inverse Functions ................ 349 7.3 Exponential and Logarithmic Functions ............. 355 7.4 Hyperbolic Functions ...................... 364 7.5 L’Hôpital’s Rule ......................... 375 8 Techniques of Integration 383 8.1 Integration by Parts ....................... 383 8.2 Trigonometric Integrals ..................... 394 8.3 Trigonometric Substitution ................... 407 8.4 Partial Fraction Decomposition ................. 416 8.5 Integration Strategies ...................... 426 8.6 Improper Integration ...................... 435 8.7 Numerical Integration ...................... 447 9 Sequences and Series 461 9.1 Sequences ............................ 461 9.2 Infinite Series .......................... 476 9.3 The Integral Test ......................... 490 9.4 Comparison Tests ........................ 496 9.5 Alternating Series and Absolute Convergence ......... 505 9.6 Ratio and Root Tests ....................... 517 9.7 Strategy for Testing Series .................... 523 9.8 Power Series ........................... 527 9.9 Taylor Polynomials ....................... 545 9.10 Taylor Series ........................... 555 10 Curves in the Plane 575 10.1 Arc Length and Surface Area .................. 575 10.2 Parametric Equations ...................... 585 10.3 Calculus and Parametric Equations ............... 597 10.4 Introduction to Polar Coordinates ............... 611 10.5 Calculus and Polar Functions .................. 624 Calculus III 11 Vectors 639 11.1 Introduction to Cartesian Coordinates in Space ........ 639 11.2 An Introduction to Vectors ................... 656 11.3 The Dot Product ......................... 670 11.4 The Cross Product ........................ 684 11.5 Lines ............................... 697 11.6 Planes .............................. 707 12 Vector Valued Functions 715 12.1 Vector‐Valued Functions .................... 715 12.2 Calculus and Vector‐Valued Functions ............. 722 12.3 The Calculus of Motion ..................... 735 12.4 Unit Tangent and Normal Vectors ................ 748 12.5 The Arc Length Parameter and Curvature ........... 758 13 Functions of Several Variables 769 13.1 Introduction to Multivariable Functions ............ 769 13.2 Limits and Continuity of Multivariable Functions ........ 777 13.3 Partial Derivatives ........................ 789 13.4 Differentiability and the Total Differential ........... 800 13.5 The Multivariable Chain Rule .................. 808 13.6 Directional Derivatives ..................... 817 13.7 Tangent Lines, Normal Lines, and Tangent Planes ....... 828 13.8 Extreme Values ......................... 839 13.9 Lagrange Multipliers ...................... 850 14 Multiple Integration 857 14.1 Iterated Integrals and Area ................... 857 14.2 Double Integration and Volume ................. 867 14.3 Double Integration with Polar Coordinates ........... 879 14.4 Center of Mass ......................... 887 14.5 Surface Area ........................... 899 14.6 Volume Between Surfaces and Triple Integration ........ 906 14.7 Triple Integration with Cylindrical and Spherical Coordinates . 928 15 Vector Analysis 945 15.1 Introduction to Line Integrals .................. 946 15.2 Vector Fields ........................... 957 15.3 Line Integrals over Vector Fields ................ 967 15.4 Flow, Flux, Green’s Theorem and the Divergence Theorem .. 980 15.5 Parameterized Surfaces and Surface Area ........... 990 15.6 Surface Integrals ......................... 1002 15.7 The Divergence Theorem and Stokes’ Theorem ........ 1011 Appendix Solutions To Selected Problems A.3 Index A.17 Important Formulas A.21 PREFACE A Note on Using this Text Thank you for reading this short preface. Allow us to share a few key points about the text so that you may better understand what you will find beyond this page. This text comprises a three‐volume series on Calculus. The first part covers material taught in many “Calculus 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6. The second text covers materi‐ al often taught in “Calculus 2”: integration and its applications, along withan introduction to sequences, series and Taylor Polynomials, found in Chapters 7 through 10. The third text covers topics common in “Calculus 3” or “Multi‐ variable Calculus”: parametric equations, polar coordinates, vector‐valued func‐ tions, and functions of more than one variable, found inChapters 11 through 15. All three are available separately for free. Printing the entire text as one volume makes for a large, heavy, cumbersome book. One can certainly only print the pages they currently need, but some prefer to have a nice, bound copy of the text. Therefore this text has been split into these three manageable parts, each of which can be purchased separately. A result of this splitting is that sometimes material is referenced that isnot contained in the present text. The context should make it clear whether the “missing” material comes before or after the current portion. Downloading the appropriate pdf, or the entire APEX Calculus LT pdf, will give access to these topics. For Students: How to Read this Text Mathematics textbooks have a reputation for being hard to read. High‐level mathematical writing often seeks to say much with few words, and thisstyle often seeps into texts of lower‐level topics. This book was written with thegoal of being easier to read than many other calculus textbooks, without becoming too verbose. Each chapter and section starts with an introduction of the coming materi‐ al, hopefully setting the stage for “why you should care,” and ends with alook ahead to see how the just‐learned material helps address future problems. Ad‐ ditionally, each chapter includes a section zero, which provides a basic review and practice problems of pre‐calculus skills. Since this content is a pre‐requisite for calculus, reviewing and mastering these skills are considered your responsi‐ bility. This means that it is your responsibility to seek assistance outside of class from your instructor, a math resource center or other math tutoring available on‐campus. A solid understanding of these skills is essential to your success in solving calculus problems. Please read the text; it is written to explain the concepts of Calculus. There are numerous examples to demonstrate the meaning of definitions, the truth of theorems, and the application of mathematical techniques. When you en‐ counter a sentence you don’t understand, read it again. If it still doesn’t make sense, read on anyway, as sometimes confusing sentences are explained by later sentences. You don’t have to read every