Active calculus 2 1

This book is different. The text is available in three different formats: HTML, PDF, and print, each of which is available via links on the landing page at https:activecalculus.orgsingle. The first two formats are free. If you are going to use the book electronically, the best mode is the HTML. The HTML version looks great in any browser, including on a smartphone, and the links are much easier to navigate in HTML than in PDF. Some particular direct suggestions about using the HTML follow among the next few paragraphs; alternatively, you can watch this short video from the author. It is also wise to download and save the PDF, since you can use the PDF offline, while the HTML version requires an internet connection. A print copy costs about 21 via Amazon. This book is intended to be read sequentially and engaged with, much more than to be used as a lookup reference. For example, each section begins with a short introduction and a Preview Activity; you should read the short introduction and complete the Preview Activity prior to class. Your instructor may require you to do this. Most Preview Activities can be completed in 1520 minutes and are intended to be accessible based on the understanding you have from preceding sections. There are not answers provided to Preview Activities, as these are designed simply to get you thinking about ideas that will be helpful in work on upcoming new material. As you use the book, think of it as a workbook, not a workedbook. There is a great deal of scholarship that shows people learn better when they interactively engage and struggle with ideas themselves, rather than passively watch others. Thus, instead of reading worked examples or watching an instructor complete examples, you will engage with Activities that prompt you to grapple with concepts and develop deep understanding. You should expect to spend time in class working with peers on Activities and getting feedback from them and from your instructor. You can purchase a separate Activities Workbook from Amazon (Chapters 14, Chapters 58) in which to record your work on the activities, or you can ask your instructor for a copy of the PDF file that has only the activities along with room to record your work. Your goal should be to do all of the activities in the relevant sections of the text and keep a careful record of your work. You can find answers to the activities in the back matter. Each section concludes with a Summary. Reading the Summary after you have read the section and worked the Activities is a good way to find a short list of key ideas that are most essential to take from the section. A good study habit is to write similar summaries in your own words

ACTIVE CALCULUS 2018 Edition - UPDATED Matthew Boelkins David Austin Steven Schlicker Active Calculus Active Calculus Matthew Boelkins Grand Valley State University Contributing Authors David Austin Grand Valley State University Steven Schlicker Grand Valley State University Production Editor Mitchel T Keller Morningside College July 2, 2019 Cover Photo: James Haefner Photography Edition: 2018 Updated Website: http://activecalculus.org ©2012–2019 Matthew Boelkins Permission is granted to copy and (re)distribute this material in any format and/or adapt it (even commercially) under the terms of the Creative Commons Attribution-ShareAlike 4.0 International License The work may be used for free in any way by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original All trademarks™ are the registered® marks of their respective owners The graphic that may appear in other locations in the text shows that the work is licensed with the Creative Commons and that the work may be used for free by any party so long as attribution is given to the author(s) and if the material is modified, the resulting contributions are distributed under the same license as this original Full details may be found by visiting https://creativecommons.org/licenses/by-sa/4.0/ or sending a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA vi Acknowledgements This text began as my sabbatical project in the winter semester of 2012, during which I wrote most of the material for the first four chapters For the sabbatical leave, I am indebted to Grand Valley State University for its support of the project, as well as to my colleagues in the Department of Mathematics and the College of Liberal Arts and Sciences for their endorsement of the project I’m also grateful to the American Institute of Mathematics for their support of free and open textbooks in general, and their support of this one in particular The beautiful full-color eps graphics in the text are only possible because of David Austin of GVSU and Bill Casselman of the University of British Columbia Building on their longstanding efforts to develop tools for high quality mathematical graphics, David wrote a library of Python routines that employ Bill’s PiScript program; David’s routines are so easy to use that even I could generate graphics like the professionals that he and Bill are I am deeply grateful to them both The current html version of the text is possible only because of the amazing work of Rob Beezer and his development of the original Mathbook XML, now known as PreTeXt My ability to take advantage of Rob’s work is due in large part to the support of the American Institute of Mathematics, which funded me to attend a weeklong workshop in Mathbook XML in San Jose, CA, in April 2016, as well as the support of the ongoing user group A subsequent workshop in June 2019 has offered further support and more improvements to the text David Farmer’s conversion script saved me hundreds of hours of work by taking my original LATEX source and converting it to PreTeXt; David remains a major source of ongoing support and advocacy Alex Jordan of Portland Community College has also been a tremendous help, and it is through Alex’s fantastic work that live WeBWorK exercises are not only possible, but also included from the 2017 version forward Mitch Keller of Morningside College agreed in early 2018 to serve as the book’s production editor; his technical expertise has contributed to many aspects of the book, including the presence of answers to activities and non- WeBWorK exercises and other supporting material for instructors For the 2018 edition, Kathy Yoshiwara of the AIM Editorial Board read the entire text and contributed editorial suggestions for every section In short, she made the prose cleaner, more direct, and simply better I’m deeply thankful for her time, effort, and insights Over my more than 20 years at GVSU, many of my colleagues have shared with me ideas and resources for teaching calculus I am particularly indebted to David Austin, Will Dickinson, Paul Fishback, Jon Hodge, and Steve Schlicker for their contributions that improved my teaching of and thinking about calculus, including materials that I have modified and used over many different semesters with students Parts of these ideas can be found throughout this text In addition, Will Dickinson and Steve Schlicker provided me access to a large number of their electronic notes and activities from teaching of differential and integral calculus, and those ideas and materials have similarly impacted my work and writing in positive ways, with some of their problems and approaches finding parallel presentation here In the summer of 2012, David and Steve each agreed to write a chapter to support the completion of the material on integral calculus David is the lead author of Chapter and Steve the lead author of Chapter Along with our colleague Ted Sundstrom, Steve has also contributed a large number of problem and activity solutions and answers I’m especially grateful for how the work of these friends and colleagues has made the text so much better Shelly Smith of GVSU and Matt Delong of Marian University both provided extensive comments on the first few chapters of early drafts, feedback that was immensely helpful in improving the text As more and more people use the text, I am grateful to everyone who reads, edits, and uses this book, and hence contributes to its improvement through ongoing discussion Finally, I am grateful for all that my students have taught me over the years Their responses and feedback have helped to shape me as a teacher, and I appreciate their willingness to wholeheartedly engage in the activities and approaches I’ve tried in class, to let me know how those affect their learning, and to help me learn and grow as an instructor Early on, they also provided useful editorial feedback on this text Any and all remaining errors or inconsistencies are mine I will gladly take reader and user feedback to correct them, along with other suggestions to improve the text viii Contributors A large and growing number of people have generously contributed to the development or improvement of the text Contributing authors David Austin and Steven Schlicker have each written drafts of at least one full chapter of the text Production editor Mitchel Keller has been an indispensable source of technological support and editorial counsel The following contributing editors have offered feedback that includes information about typographical errors or suggestions to improve the exposition David Austin GVSU Rene Ardila GVSU Patti Hunter Westmont College Mitchel Keller Morningside College Allan Bickle GVSU Sam Kolins Lebanon Valley College David Clark GVSU Will Dickinson Dave Kung St Mary’s College of Maryland GVSU Nate Eldredge University of Northern Colorado Charles Fortin Champlain Regional College St-Lambert, Quebec, Canada Marcia Frobish GVSU Teresa Gonske University of Northwestern - St Paul Paul Latiolais Portland State University Hugh McGuire GVSU Martin Mohlenkamp Ohio University Ray Rosentrater Westmont College Luis Sanjuan Conservatorio Profesional de Musica de Avila Spain Michael Santana GVSU Steven Schlicker GVSU Amy Stone GVSU Robert Talbert GVSU Greg Thull GVSU Michael Shulman University of San Diego Sue Van Hattum Contra Costa College Brian Stanley Foothill Community College Kathy Yoshiwara x AIM Editorial Board 7.3.4.5 Answer a K 1.054; y(1) 2.6991 b K 1.272; y(1) 2.7169 c K 0.122 and y(0.3) 0.0412 7.3.4.6 Answer a y(1) ≈ y5 2.7027 b y(1) ≈ y10 2.7141 c The square of ∆t 7.4 · Separable differential equations 7.4.3.6 Answer a dM dt kM b M(t) M0 e kt c M(t) M0 e − 5730 t ≈ M0 e −0.000121t ln(2) d t 5730 ln(4) ln(2) e t − ≈ 11460 years 5730 ln(0.3) ln(2) ≈ 9952.8 years 7.4.3.7 Answer √ a y 64 − t b −8 ≤ t ≤ c y(8) d dy dt − yt is not defined when y 7.4.3.8 Answer √ a dh h k dt b The tank with k c k d h(t) −10 has water leaving the tank much more rapidly −5 (5 − 2.5t)2 e 2.5 minutes f No 7.4.3.9 Answer (a) 613 Appendix C Answers to Selected Exercises (b) P is stable (c) P(t) 3e ln( )e (d) P(t) 3e ln(2)e −t −t (e) Yes 7.5 · Modeling with differential equations 7.5.3.6 Answer a + 0.05A dA dt b A(25) 49.80686 million dollars c A(25) 34.90343 million dollars d The first e t 20 ln(2) ≈ 13.86 years 7.5.3.7 Answer a 9.8 − kv dv dt 9.8 k b v is a stable equilibrium 9.8−9.8e −kt k c v(t) d k 9.8/54 ≈ 0.181481 e t ln(0.5) −0.181481 ≈ 3.1894 seconds 7.5.3.8 Answer a dw dt b w(t) k w √ 7t + 64; w(12) c The model is unrealistic 614 √ 148 ≈ 12.17 pounds 7.5.3.9 Answer a The inflow and outflow are at the same rate b 60 grams per minute c S(t) grams 100 gallon d 3S(t) grams 100 minute e dS dt 60 − f S 100 S 2000 is a stable equilibrium solution g S(t) 2000 − 2000e − 100 t h S(t) → 2000 7.6 · Population Growth and the Logistic Equation 7.6.4.5 Answer a p(t) → as t → ∞ provided p(0) > b p(t) 9e −0.2t +1 c t −5 ln(1/9) ≈ 10.986 days d t −5 ln(0.2/9) ≈ 19.033 days 7.6.4.6 Answer a db dt b b 3000 b(15000 15000 c When b d t − b) 7500 − 15 ln(1/70) ≈ 0.8497 days 7.6.4.7 Answer a 10000 fish b dP dt 0.1P(10 − P) − 0.2P c 8000 fish d P(1) ≈ 8.7899 thousand fish e t −1.2 ln(5/11) ≈ 0.9461 years · Sequences and Series 8.1 · Sequences 8.1.3.5 Answer a Unclear whether it converges or diverges 615 Appendix C Answers to Selected Exercises n ln(n) n t ln(n) n b If limx→∞ f (x) 0.2987 L, then limn→∞ c lim n→∞ 0.3466 0.2781 ln(n) n ln(n) n 0.3663 0.2599 0.3466 0.2442 0.3218 10 0.2303 L as well lim x→∞ ln(x) x 8.1.3.6 Answer a P1 (r) 12 in interest in the second month; at the end of the second month, P2 ( b P3 P 1+ c Pn P 1+ ( ) r 12 ) r n 12 ( P 1+ ) r 12 is a pattern to these calculations 8.1.3.7 Answer a A1 b A2 c A3 d A4 e A n (100) ( )2 ( )3 ( )4 · 100 · 100 · 100 · 100 ( )n f A n → as time goes on g It takes about 6.6439 half-lives to elapse to get down to gram remaining, or 5·6.6439 33.2193 minutes 8.1.3.8 Answer a The data points not appear periodic at all b At least 13 samples, so at least every 10/13 ≈ 0.76923 seconds c 44100 Hz is slightly more than double 20 KHz 8.2 · Geometric Series 616 8.2.3.5 Answer a 30 · 500 1500 dollars b ( c $0.01 230 − Day 10 ) Pay on this day $0.01 $0.02 $0.04 $0.08 $0.16 $0.32 $0.64 $1.28 $2.56 $5.12 Total amount paid to date $0.01 $0.03 $0.07 $0.15 $0.31 $0.63 $1.27 $2.55 $5.11 $10.23 $10, 737, 418.23 8.2.3.6 Answer a h1 b h2 c h3 d h n (3) (3) (3) h h1 h2 h n−1 (3) ( )2 ( )3 h h ( )n h e The distance traveled by the ball is 7h, which is finite 8.2.3.7 Answer a There are equally possible outcomes when we roll one die b The three rolls are independent so the probability of the overall outcome is the product of the three probabilities c See (b) d P 11 8.2.3.8 Answer a 0.75P dollars spent b 0.75P + 0.75(0.75P) 0.75P(1 + 0.75) dollars c 0.75P + 0.752 P + 0.752 P + · · · 0.75P(1 + 0.75 + 0.752 + · · · ) dollars d A stimulus of 200 billion dollars adds 600 billion dollars to the economy 8.2.3.9 Answer a (r) 12 P1 dollars 617 Appendix C Answers to Selected Exercises ( b P2 ( c P2 ( d P3 ( P3 1+ r 12 P1 − M 1+ ) r 12 1+ r 12 1+ r 12 ( P 1+ e Pn ) ) g A(t) [ ( P− 1+ 1+ ) r n 12 ( ( ( P2 − M )3 − ) r 12t 12 P 1+ f P(t) [ P− 1+ 1+ 1000 − ( 12M ) ( ( r − r 12 r )] ) 1+ ( 12M ) ( ( ) 12(25) ( 0.2 r 12 ( + 1+ ) r n 12 1+ ) 0.2 12t 12 1+ M ) r 12 M ) −1 ) r 12t 12 + ] −1 ) 12(25) 0.2 t ≈ 5.5 We pay $659 dollars in interest on our $1000 loan h $291.74 each month to complete the loan in years; we pay $2,504.40 in interest 8.3 · Series of Real Numbers 8.3.7.5 Answer a ∑ 10k b ∑ 10k k! k! converges converges c The sequence { bn } n! has to converge to 8.3.7.6 Answer √ a n a n ≈ r for large n b a n+1 an ≈ r c < r < 8.3.7.7 Answer a i a n + 2n →1 and b n −1 → −1 ii The series is geometric with r iii Since the two individual series diverge, neither sum is a finite number, so it doesn’t make any sense to add them b i Note that A n + B n ii Note that limn→∞ c ∑∞ k 2k +3k 5k 25 8.3.7.8 Answer a i) S1 and T1 ii) S2 > T2 iii) S3 > T3 618 (a1 + b ) + (a + b2 ) + · · · + (a n + b n ) ∑n k (a k + bk ) limn→∞ (∑n k ak + ∑n k ) bk iv) S n > Tn v) ∑ ∑ > k2 ∑ ; k +k ∑ k +k converges ∑ b If b k diverges, then b k is infinite, and anything larger must also be infinite; if is convergent then anything smaller and positive must also be finite i) Note that < ii) Note that k3 k > < ∑ ak k−1 k +1 8.4 · Alternating Series 8.4.6.5 Answer a ∑∞ k 2k+1 b |S100 − diverges by comparison to the Harmonic series ∑∞ k k (−1) 2k+1 | 4,999,999,999.5 8.4.6.6 Answer a S n +S n+1 S n +S n +(−1)n+2 a n+1 b S20 0.668771403 ; 0.0006 8.4.6.7 Answer a {1} n { S20 +S21 161227687 232792560 0.692580926 , accurate to within about } and − n12 converge to b Notice that k − k2 k−1 k2 and compare to the Harmonic series c It is possible for a series to alternate, have the terms go to zero, have the terms not decrease to zero, and the series diverge 8.5 · Taylor Polynomials and Taylor Series 8.5.6.6 Answer a P3 (x) −1 + 3x − b For n ≥ 3, Pn (x) c For n ≥ m, Pn (x) 2! x + 3! x , which is the same polynomial as f (x) f (x) f (x) 8.5.6.7 Answer a P1 (x) P2 (x) P3 (x) ( π) ( π) 1+0 x− − ( ) π 1+0 x− − ( π )2 1− x− 2! 1+0 x− 1 2! 2! ( π )2 ( π )2 1− x− 2! ( π )2 ( π )3 x− + x− 3! x− P2 (x) 619 Appendix C Answers to Selected Exercises ( P4 (x) 2! 1− 2! 1− P(x) b (x − 1)2 + (x − 1)3 − (x − 1)4 2! 3! 4! 1 1 1(x − 1) − (x − 1)2 + (x − 1)3 − (x − 1)4 + (x − 1)5 − · · · P4 (x) P(x) π) ( π )2 ( π )3 ( π )4 − x− x− x− + + 2! 3! 4! ( π )2 ( π )4 x− + x− 4! ( ) ( π π )4 ( π )6 x− + x− − x− +··· 4! 6! 1+0 x− + 1(x − 1) − c P101 (1) ≈ 0.698073 8.5.6.8 Answer a P4 (x) b i x2 (x) P(x ) x2 − 3! x + 1 5! x − · · · ii All real numbers 8.6 · Power Series 8.6.4.3 Answer a sin(x ) b c ∫ k x 2(2k+1) k (−1) (2k+1)! , sin(x ) dx ∫1 ∑∞ sin(x ) dx with interval of convergence (−∞, ∞) ∑∞ x 4k+3 k k (−1) (2k+1)!(4k+3) ∑∞ + C k k (−1) (2k+1)!(4k+3) Use n to generate the desired estimate 8.6.4.4 Answer a b Then f ′(x) ∞ ∑ ka k x k−1 k f ′′(x) ∞ ∑ k(k − 1)a k x k−2 k f (3) (x) ∞ ∑ k(k − 1)(k − 2)a k x k−3 k f (n) (x) ∞ ∑ k n 620 k(k − 1)(k − 2) · · · (k − n + 1)a k x k−n So f (0) ′ f (0) ′′ f (0) (3) a0 a1 2!a (0) 3!a f (k) (0) k!a k f and f (k) (0) k! for each k ≥ But these are just the coefficients of the Taylor series expansion of f , which leads us to the following observation ak 8.6.4.6 Answer The results from the various part of this exercise show that ( y a0 + ( ∞ ∑ x 3k (2)(3)(5)(6) · · · (3k − 1)(3k) k ∞ ∑ + a1 x + k ) ) x 3k+1 (3)(4)(6)(7) · · · (3k)(3k + 1) 621 Appendix C Answers to Selected Exercises 622 Index u-substitution, 284 concave up, 60 concavity, 59 conditional convergence, 475 constant multiple rule, 92 continuous, 72 continuous at x a, 72 converge sequence, 438 convergence absolute, 475 conditional, 475 convergent sequence, 438 cosecant, 116 cotangent, 116 critical number, 159 critical point, 159 critical value, 159 cusp, 75 absolute convergence, 475 acceleration, 60 alternating series, 471 alternating series estimation theorem, 473 alternating series test, 473 antiderivative, 206 general, 249, 263 graph, 261 antidifferentiation, 205 arc length, 330, 331 arcsine, 133 area, 326 under velocity function, 203 asymptote, 151 horizontal, 151 vertical, 151 autonomous, 382 average rate of change, 22 average value, 238 average value of a function, 237 average velocity, 2, decreasing, 54 definite integral definition, 230, 231 density, 345 derivative arcsine, 134 constant function, 91 cosine, 101 cotangent, 117 definition, 23, 36 exponential function, 91 inverse, 136 logarithm, 131 power function, 91 sine, 101 tangent, 117 difference quotient, 47 backward difference, 47 carrying capacity, 427 center of mass (continuous mass distribution), 350 center of mass (point-masses, 350 central difference, 47 chain rule, 122 codomain, 130 composition, 120 concave down, 60 623 Index differentiable, 24, 73 differential equation, 376 autonomous, 382 first order, 382 solution, 379 disk method, 336 distance traveled, 205 diverge sequence, 438 Divergence Test, 459 domain, 130 dominates, 156 equilibrium solution, 390 stable, 390 unstable, 390 error, 313 error function, 274, 306 Euler’s Method, 398 error, 402 extreme value, 158 extreme value theorem, 180 Fibonacci sequence, 435 finite geometric series, 444 first derivative test, 159, 160 foot-pound, 355 forward difference, 47 FTC, 247 function, 130 function-derivative pair, 284 Fundamental Theorem of Calculus First, 270 Second, 272 fundamental theorem of calculus, 245, 247 geometric series, 445 common ratio, 445 harmonic series, 459 Hooke’s Law, 357 implicit function, 140 improper integral, 366 converges, 368 diverges, 368 unbounded integrand, 368 624 unbounded region of integration, 367 increasing, 54 indefinite integral, 283 evaluate, 283 indeterminate form, 14 infinite series, 455 infinity, 150 inflection point, 164 initial condition, 263 instantaneous rate of change, 23, 46 instantaneous velocity, 3, 17 integral function, 264 integral sign, 231 integral test, 459, 462 integrand, 231 integration by parts, 292 interval of convergence, 490 left limit, 69 lemniscate, 140 limit definition, 13 one-sided, 69 limit comparison test, 462 limits of integration, 231 local linearization, 81 locally linear, 74 logistic, 427 logistic equation, 427 solution, 429 Maclaurin series, 489 mass, 345 maximum absolute, 157 global, 157 local, 157 relative, 157 midpoint rule error, 313 minimum absolute, 157 global, 157 local, 157 relative, 157 net signed area, 222 Newton’s Law of Cooling, 383 Newton-meter, 355 one-to-one, 130 onto, 130 partial fractions, 302 partial sum, 446, 456 per capita growth rate, 425 position, power series, 499 definition, 499 product rule, 106 quotient rule, 108 ratio test, 464 related rates, 193 Riemann sum, 219 left, 219 middle, 220 right, 219 right limit, 69 secant, 116 secant line, 24 second derivative, 57 second derivative test, 162 second fundamental theorem of calculus, 273 separable, 407, 408 sequence, 437 term, 437 sequence of partial sums, 456 series converges, 457 diverges, 457 geometric, 445 sigma notation, 217 Simpson’s rule, 316 slope field, 387, 388 solid of revolution, 335 stable, 390 sum rule, 93 tangent, 115 tangent line, 24 equation, 81 Taylor polynomial, 486 error, 492 Taylor polynomials, 484 Taylor series, 488, 489 interval of convergence, 490 radius of convergence, 492 total change theorem, 250, 251 trapezoid rule, 312 error, 313 triangular numbers, 435 trigonometry, 115 fundamental trigonometric identity, 115 unstable, 390 washer method, 337 weighted average, 316, 348 work, 355, 356 625 Index 626 Colophon This book was authored in PreTeXt .. .Active Calculus Active Calculus Matthew Boelkins Grand Valley State University Contributing Authors David Austin... Yoshiwara x AIM Editorial Board Active Calculus: Our Goals Several fundamental ideas in calculus are more than 2000 years old As a formal subdiscipline of mathematics, calculus was first introduced... to solve Calculus belongs to humankind, not any individual author or publishing company Thus, a primary purpose of this work is to present a calculus text that is free See https://activecalculus.org