Calculus workbook for dummies

Calculus Workbook For Dummies®, 3rd Edition with Online Practice Published by: John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, www.wiley.com Copyright © 2018 by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the Publisher Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions Trademarks: Wiley, For Dummies, the Dummies Man logo, Dummies.com, Making Everything Easier, and related trade dress are trademarks or registered trademarks of John Wiley & Sons, Inc., and may not be used without written permission All other trademarks are the property of their respective owners John Wiley & Sons, Inc., is not associated with any product or vendor mentioned in this book LIMIT OF LIABILITY/DISCLAIMER OF WARRANTY: THE PUBLISHER AND THE AUTHOR MAKE NO REPRESENTATIONS OR WARRANTIES WITH RESPECT TO THE ACCURACY OR COMPLETENESS OF THE CONTENTS OF THIS WORK AND SPECIFICALLY DISCLAIM ALL WARRANTIES, INCLUDING WITHOUT LIMITATION WARRANTIES OF FITNESS FOR A PARTICULAR PURPOSE NO WARRANTY MAY BE CREATED OR EXTENDED BY SALES OR PROMOTIONAL MATERIALS THE ADVICE AND STRATEGIES CONTAINED HEREIN MAY NOT BE SUITABLE FOR EVERY SITUATION THIS WORK IS SOLD WITH THE UNDERSTANDING THAT THE PUBLISHER IS NOT ENGAGED IN RENDERING LEGAL, ACCOUNTING, OR OTHER PROFESSIONAL SERVICES IF PROFESSIONAL ASSISTANCE IS REQUIRED, THE SERVICES OF A COMPETENT PROFESSIONAL PERSON SHOULD BE SOUGHT NEITHER THE PUBLISHER NOR THE AUTHOR SHALL BE LIABLE FOR DAMAGES ARISING HEREFROM THE FACT THAT AN ORGANIZATION OR WEBSITE IS REFERRED TO IN THIS WORK AS A CITATION AND/OR A POTENTIAL SOURCE OF FURTHER INFORMATION DOES NOT MEAN THAT THE AUTHOR OR THE PUBLISHER ENDORSES THE INFORMATION THE ORGANIZATION OR WEBSITE MAY PROVIDE OR RECOMMENDATIONS IT MAY MAKE FURTHER, READERS SHOULD BE AWARE THAT INTERNET WEBSITES LISTED IN THIS WORK MAY HAVE CHANGED OR DISAPPEARED BETWEEN WHEN THIS WORK WAS WRITTEN AND WHEN IT IS READ For general information on our other products and services, please contact our Customer Care Department within the U.S at 877-762-2974, outside the U.S at 317-572-3993, or fax 317-572-4002 For technical support, please visit www.wiley.com/techsupport Wiley publishes in a variety of print and electronic formats and by printon-demand Some material included with standard print versions of this book may not be included in e-books or in print-on-demand If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Library of Congress Control Number: 2018933792 ISBN 978-1-119-35748-3 (pbk); ISBN 978-1-119-35750-6 (ebk); ISBN 978-1-119-35749-0 Calculus Workbook For Dummies® To view this book's Cheat Sheet, simply go to www.dummies.com and search for “Calculus Workbook For Dummies Cheat Sheet” in the Search box Table of Contents Cover Introduction About This Book Foolish Assumptions Icons Used in This Book Beyond the Book Where to Go from Here Part 1: Pre-Calculus Review Chapter 1: Getting Down to Basics: Algebra and Geometry Fraction Frustration Misc Algebra: You Know, Like Miss South Carolina Geometry: When Am I Ever Going to Need It? Solutions for This Easy, Elementary Stuff Chapter 2: Funky Functions and Tricky Trig Figuring Out Your Functions Trigonometric Calisthenics Solutions to Functions and Trigonometry Part 2: Limits and Continuity Chapter 3: A Graph Is Worth a Thousand Words: Limits and Continuity Digesting the Definitions: Limit and Continuity Taking a Closer Look: Limit and Continuity Graphs Solutions for Limits and Continuity Chapter 4: Nitty-Gritty Limit Problems Solving Limits with Algebra Pulling Out Your Calculator: Useful “Cheating” Making Yourself a Limit Sandwich Into the Great Beyond: Limits at Infinity Solutions for Problems with Limits Part 3: Differentiation Chapter 5: Getting the Big Picture: Differentiation Basics The Derivative: A Fancy Calculus Word for Slope and Rate The Handy-Dandy Difference Quotient Solutions for Differentiation Basics Chapter 6: Rules, Rules, Rules: The Differentiation Handbook Rules for Beginners Giving It Up for the Product and Quotient Rules Linking Up with the Chain Rule What to Do with Y’s: Implicit Differentiation Getting High on Calculus: Higher Order Derivatives Solutions for Differentiation Problems Chapter 7: Analy zing Those Shapely Curves with the Derivative The First Derivative Test and Local Extrema The Second Derivative Test and Local Extrema Finding Mount Everest: Absolute Extrema Smiles and Frowns: Concavity and Inflection Points The Mean Value Theorem: Go Ahead, Make My Day Solutions for Derivatives and Shapes of Curves Chapter 8: Using Differentiation to Solve Practical Problems Optimization Problems: From Soup to Nuts Problematic Relationships: Related Rates A Day at the Races: Position, Velocity, and Acceleration Solutions to Differentiation Problem Solving Chapter 9: Even More Practical Applications of Differentiation Make Sure You Know Your Lines: Tangents and Normals Looking Smart with Linear Approximation Calculus in the Real World: Business and Economics Solutions to Differentiation Problem Solving Part 4: Integration and Infinite Series Chapter 10: Getting into Integration Adding Up the Area of Rectangles: Kid Stuff Sigma Notation and Riemann Sums: Geek Stuff Close Isn’t Good Enough: The Definite Integral and Exact Area Finding Area with the Trapezoid Rule and Simpson’s Rule Solutions to Getting into Integration Chapter 11: Integration: Reverse Differentiation The Absolutely Atrocious and Annoying Area Function Sound the Trumpets: The Fundamental Theorem of Calculus Finding Antiderivatives: The Guess-and-Check Method The Substitution Method: Pulling the Switcheroo Solutions to Reverse Differentiation Problems Chapter 12: Integration Rules for Calculus Connoisseurs Integration by Parts: Here’s How u du It Transfiguring Trigonometric Integrals Trigonometric Substitution: It’s Your Lucky Day! Partaking of Partial Fractions Solutions for Integration Rules Chapter 13: Who Needs Freud? Using the Integral to Solve Your Problems Finding a Function’s Average Value Finding the Area between Curves Volumes of Weird Solids: No, You’re Never Going to Need This Arc Length and Surfaces of Revolution Solutions to Integration Application Problems Chapter 14: Infinite (Sort of) Integrals Getting Your Hopes Up with L’Hôpital’s Rule 10 11 Disciplining Those Improper Integrals Solutions to Infinite (Sort of) Integrals Chapter 15: Infinite Series: Welcome to the Outer Limits The Nifty nth Term Test Testing Three Basic Series Apples and Oranges … and Guavas: Three Comparison Tests Ratiocinating the Two “R” Tests He Loves Me, He Loves Me Not: Alternating Series Solutions to Infinite Series Part 5: The Part of Tens Chapter 16: Ten Things about Limits, Continuity, and Infinite Series The 33333 Mnemonic The 13231 Mnemonic Chapter 17: Ten Things You Better Remember about Differentiation The Difference Quotient The First Derivative Is a Rate The First Derivative Is a Slope Extrema, Sign Changes, and the First Derivative The Second Derivative and Concavity Inflection Points and Sign Changes in the Second Derivative The Product Rule The Quotient Rule Linear Approximation 10 “PSST,” Here’s a Good Way to Remember the Derivatives of Trig Functions About the Author Connect with Dummies Index End User License Agreement Pages 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 i ii 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 10 11 12 chain rule See also implicit differentiation derivative of a constant, 89 higher order derivatives, 101–102 implicit differentiation, 98–100 See also chain rule power, 89 problems, 90–102 product, 92–94 10 quotient, 92–94 11 solutions, 103–115 13 solutions, 157–172, 183–190 14 speed, 154 15 speed and distance travelled, 153–156 16 tangents, 173–176 17 velocity, 153–156 direct comparison test, 291–294, 314 disk/washer method, 258–264 displacement, 153 distance, 153 divergence nth term test of, 313 testing for, 289–298 divergent series, 287 E economics, 179 estimation area under curves, 193–195, 202–205 linear approximation, 177–179, 317 exact area, 200–201 expressions with trigonometric functions, 232–234 F factoring method, 54–59 first derivative test local extrema, 117–120, 316 rates, 315–316 sign changes, 316 slopes, 316 FOILing method, 54–59 fractions difference quotient, partial, 237–240 review problems, solutions, 16 functions See also derivatives; graphs argument, 214, 216 average value of, 255–256 continuity definition, 44 problems, 47–49 solutions, 50–52 definition, 25 horizontal asymptote, 63 identifying graph curve as, 25 length along, 265–267 limits definition, 44 problems, 47–49 solutions, 50–52 product of two, 289–290 10 quotient of two, 289–290 11 rational, 63 12 review problems, 26–28 solutions, 33–36 13 selecting, 230–233 14 vertical line test, 25 Fundamental Theorem of Calculus, 216–219 G geometric series, 289–290 geometry review problems, 11–15 solutions, 20–23 graphs See also functions continuity definition, 44 problems, 47–49 solutions, 50–52 of curves identifying as function, 25 review, 26–28 vertical line test, 25 limits definition, 44 problems, 47–49 solutions, 50–52 guess-and-check method, 216–219 H higher order derivatives, 101–102 highest/lowest points, curves, 122–125 horizontal asymptote, 63 horizontal inflection point, 316 hypotenuse, 235 I implicit differentiation, 98–100 improper integrals, 280–282 index of summation, 196 infinite discontinuity, 312 infinite integrals improper integrals, 280–282 L’Hôpital’s Rule, 277–279 solutions, 283–286 infinite limits of integration, 280–282 infinite series, 291–294 13231 mnemonic, 313–314 33333 mnemonic, 311–313 absolute convergence, 317 alternating series, 297–298 comparison tests, 291–294, 314 conditional convergence, 297 convergent series, 287 definition, 287 direct comparison test, 291–294, 314 10 divergent series, 287 11 geometric series, 289–290 12 integral comparison test, 291–294, 314 13 limit comparison test, 291–294, 314 14 nth term test, 287–288, 313 10 11 12 13 15 problems, 288–298 16 p-series, 289–290 17 ratio test, 295–297, 314 18 root test, 295–297, 314 19 sequence, 287 20 solutions, 298–307 21 telescoping series, 289–290 22 testing for divergence/convergence, 289–298 infinity, limits at, 63–66 inflection points curves, 126–129 second derivative test, 316 integral comparison test, 291–294, 314 integrals arc length, 265–267 areas between curves, 256–258, 265–267 average value of functions, 255–256 cylindrical shell method, 258–264 definite, 200–201 disk/washer method, 258–264 improper, 280–282 infinite limit of, 280–282 irregular solids, 258–264 10 length along, 265–267 11 meat slicer method, 258–265 12 problems, 256–267 13 solutions, 268–275 14 surfaces of revolution, 265–267 15 volume calculation, 258–264 integrands, 200 integration arc length, 265–267 areas See also antiderivatives under curves, 193–195, 202–205 between curves, 256–258 definite integral, 200–201 exact area, 200–201 index of summation, 196 infinite limits of, 280–282 integrands, 200 irregular shapes, 193–195 length along function, 265–267 10 long sums, shorthand for, 196–200 11 problems, 194–205, 256–267 12 rectangles, 193–195 13 Riemann sums, 196–200 14 sigma notation, 196–200 15 Simpson's rule, 202–205 16 solutions, 205–212, 268–275 17 surfaces of revolution, 265–267 18 Trapezoid rule, 202–205 cylindrical shell method, 258–264 disk/washer method, 258–264 functions, average value of, 255–256 meat slicer method, 258–264 by parts See also product rule definition, 229 expressions with trigonometric functions, 232–234 LIATE mnemonic, 230 partial fractions, 237–240 problems, 230–240 Pythagorean Theorem, 235–237 solutions, 241–254 trigonometric substitution, 235–237 problems, 256–267 solutions, 268–275 10 volumes, 258–264 K Kasube, Herbert, 230 L least common denominator method, 54–59 L’Hôpital’s Rule, 277–279 LIATE mnemonic, 230 limit comparison test, 291–294, 314 limit problems limits at infinity, 63–66 solutions, 67–76 solving with algebra, 54–59 with calculator, 59–61 canceling method, 54–59 conjugate multiplication method, 54–59 factoring method, 54–59 FOILing method, 54–59 least common denominator method, 54–59 sandwich method, 61–64 simplification method, 54–59 10 squeeze method, 61–64 limits 13231 mnemonic, 313–314 33333 mnemonic, 311–313 definition, 44 at infinity, 63–66 10 11 12 13 memorizing, 55–56 review problems, 47–49 solutions, 50–52 linear approximation, 177–179, 317 lines normal, 173–176 straight, 25 tangent, 173–176 local extrema first derivative test, 117–120, 316 second derivative test, 120–122 local max, 316 local min, 316 long sums, shorthand for, 196–200 lowest/highest points, curves, 122–125 M N marginal, 179–181 marginal profit, 179–181 marginal revenue, 179–181 Mean Value Theorem, 129–130 meat slicer method, 258–264 mnemonics 13231, 313–314 33333, 311–313 continuity, 311–314 limits, 311–314 PSST, 317 negative acceleration, 154 negative area, 214 negative second derivative, 316 negative velocity, 153, 154 normals, 173–176 nth term test, 287–288, 313 O 13231 mnemonic, 313–314 optimization, 147–149 P 10 11 12 Q palameter, 153 parallelogram, area calculation, 13 partial fractions, 237–240 position, 153–156 positive acceleration, 154 positive second derivative, 316 positive velocity, 154 power rule, 89 product rule, 92–94, 317 See also integration, by parts p-series, 289–290 PSST mnemonic, 317 Pythagorean Theorem, 235–237 quotient rule, 92–94, 317 R radian/degree conversions, 13 rates See also derivatives calculating average, 129–130 first derivative test, 315–316 related, 150–152 ratio test, 295–297, 314 rational function, 63 rectangles, area of, 193–195 related rates, 150–152 removable discontinuity, 312 reverse differentiation See antiderivatives Riemann sums, 196–200 10 right triangles, 235–237 11 root test, 295–297, 314 12 rules for differentiation beginners, 89–91 chain, 94–98 See also implicit differentiation derivative of a constant, 89 higher order derivatives, 101–102 implicit differentiation, 98–100 See also chain rule power, 89 problems, 90–102 product, 92–94 quotient, 92–94 10 solutions, 103–115 L’Hôpital’s, 277–279 Simpson's, 202–205 Trapezoid, 202–205 S sandwich method, 61–64 second derivative test concavity, 316 inflection points, 316 local extrema, 120–122 negative, 316 positive, 316 sign changes, 316 sequence, 287 series, 287 divergent series, 287 sigma notation, 196–200 sign changes first derivative test, 316 second derivative test, 316 simplification method, 54–59 Simpson's Rule, 202–205 slope calculating average, 129–130 first derivative test, 316 10 SohCahToa right triangle, 235–237 11 speed, average, 154 12 squeeze method, 61–64 13 straight lines, 25 14 substitution method, 221–224 15 surfaces of revolution, 265–267 T tangents, 173–176 telescoping series, 289–290 33333 mnemonic, 311–313 total displacement, 154 Trapezoid Rule, 202–205 triangles area, 11–12 geometry review, 11–15 hypotenuse, 235 Pythagorean Theorem, 235–237 right, 235–237 trigonometric substitution, 235–237 trigonometry review problems, 29–32 solutions, 36–40 V velocity, 153–156 vertical line test, 25 vertical tangent, 313, 316 volumes, 258–264 Take Dummies with you everywhere you go! Go to our Website Like us on Facebook Follow us on Twitter Watch us on YouTube Join us on LinkedIn Pin us on Pinterest Circle us on google+ Subscribe to our newsletter Create your own Dummies book cover Shop Online WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ... (ebk); ISBN 978-1-119-35749-0 Calculus Workbook For Dummies? ? To view this book's Cheat Sheet, simply go to www .dummies. com and search for ? ?Calculus Workbook For Dummies Cheat Sheet” in the Search... covers topics in greater depth Calculus Workbook For Dummies, 3rd Edition, gives you the opportunity to master the calculus topics you study in class or in Calculus For Dummies through a couple hundred... my other classic, Calculus For Dummies (also published by Wiley) In both books, you find calculus explained in plain English with a minimum of technical jargon Calculus For Dummies covers topics