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Unformatted text preview: with Open Texts CALCULUS Early Transcendentals an Open Text Adapted for ATHABASCA UNIVERSITY MATH 266 – CALCULUS II NOVEMBER 2017 EDITION 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 ONLINE ASSESSMENT OPEN TEXT All digital forms of access to our high-quality open texts are entirely FREE! All content is reviewed for excellence and is wholly adaptable; custom editions are produced by Lyryx for those adopting Lyryx assessment. Access to the original source files is also open to anyone! We have been developing superior online formative assessment for more than 15 years. Our questions are continuously adapted with the content and reviewed for quality and sound pedagogy. To enhance learning, students receive immediate personalized feedback. Student grade reports and performance statistics are also provided. INSTRUCTOR SUPPLEMENTS SUPPORT Access to our in-house support team is available 7 days/week to provide prompt resolution to both student and instructor inquiries. In addition, we work one-on-one with instructors to provide a comprehensive system, customized for their course. This can include adapting the text, managing multiple sections, and more! Additional instructor resources are also freely accessible. Product dependent, these supplements include: full sets of adaptable slides and lecture notes, solutions manuals, and multiple choice question banks with an exam building tool. Contact Lyryx Today! [email protected] 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 [email protected] with your ideas. 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 Peter Chow Nathan Friess Stephanie Keyowski Claude Laflamme Martha Laflamme Jennifer MacKenzie Tamsyn Murnaghan Bogdan Sava Ryan Yee 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 B Extensive edits, additions, and revisions have been completed by the editorial team at Lyryx Learning. All new content (text and images) is released under the same license as noted above. Adapted for Athabasca University, October 2017 • G. Hartman: – Many new exercises are included, adapted by Lyryx from APEX Calculus. The following exercises are from APEX: 1.2.2 to 1.2.21, 3.1.15 to 3.1.56, 3.2.13 to 3.2.35, 3.3.11 to 3.3.22, 3.4.11 to 3.4.33, 3.7.11 to 3.7.37, 4.2.10 to 4.2.26 – New content on Hyperbolic Functions 1.7 and Inverse Hyperbolic Functions 2.2 is included. These sections were adapted by Lyryx from the section “Hyperbolic Functions” in APEX Calculus. – APEX Calculus: Version 3.0 written by G. Hartman. T. Siemers and D. Chalishajar of the Virginia Military Instititue and B. Heinold of Mount Saint Mary’s University also contributed to APEX Calculus. This material is released under Creative Commons license CC BY-NC ( ). See for more information and original version. • OpenStax: New content is included; Table of Integrals in Additional Material. This section was adapted by Lyryx from the section of the same name in Calculus Volume 1 by OpenStax. This material is released under Creative Commons license CC BY-NCDownload for free at SA ( ). ( ). • Lyryx: – Front matter has been updated including cover page, copyright, and revision pages. 2017 A – Several examples and exercises from Chapter 15 and 16 have been rewritten or removed. – 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 Calculus chapters. 2016 B • Lyryx: Exercise numbering has been updated to restart with each section. • G. Hartman: New content on Riemann Sums 3.5 is included. This section was adapted by Lyryx from the section of the same name in APEX Calculus 3.0. See above for APEX Calculus license information. Continued on next page ... ii 2016 A • Lyryx: The layout and appearance of the text has been updated, including the title page and newly designed back cover. 2015 A • J. Ling: Addition of new exercises and proofs throughout. Revised arrangement of topics in the Application of Derivatives chapter. Continuity section revised to include additional explanations of content and additional examples. • M. Cavers: Addition of new material and images particularly in the Review chapter. 2014 A 2012 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 modification and expansion of notes written and released by N. Koblitz of the University of Washington. That version also contains exercises and examples from Elementary Calculus: An Approach Using Infinitesimals, written by H. J. Keisler of the University of Wisconsin under a Creative Commons license (see ). A. Schueller, B. Balof, and M. Wills all of Whitman College, have also contributed content. This material is released under the Creative Commons Attribution-NonCommercialSee ShareAlike License ( ). for more information. Contents Contents iii Introduction and Review 1 Unit 1: 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Inverse Functions Inverse Functions . . . . . . . . . . . . . . . . . . . Derivatives of Inverse Functions . . . . . . . . . . . Exponential Functions . . . . . . . . . . . . . . . . Logarithms . . . . . . . . . . . . . . . . . . . . . . Derivatives of Exponential & Logarithmic Functions Logarithmic Differentiation . . . . . . . . . . . . . . Hyperbolic Functions . . . . . . . . . . . . . . . . . Unit 2: 2.1 2.2 2.3 2.4 Inverse Trigonometric and Hyperbolic Functions; L’Hopital’s Rule Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unit 3: 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Techniques of Integration Integration by Parts . . . . . . . . . Powers of Trigonometric Functions . Trigonometric Substitutions . . . . Rational Functions . . . . . . . . . Riemann Sums . . . . . . . . . . . Numerical Integration . . . . . . . . Improper Integrals . . . . . . . . . Additional Exercises . . . . . . . . Unit 4: 4.1 4.2 4.3 4.4 Applications of Integration Volume . . . . . . . . . . Arc Length . . . . . . . . Surface Area . . . . . . . Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 5 8 11 15 20 22 . . . . 27 27 33 38 39 . . . . . . . . 45 45 50 59 67 72 87 91 101 . . . . 103 103 109 112 116 iv Contents Unit 5: 5.1 5.2 5.3 5.4 Differential Equations First Order Differential Equations . . . . . First Order Homogeneous Linear Equations First Order Linear Equations . . . . . . . . Approximation . . . . . . . . . . . . . . . Unit 6: Sequences and Infinite Series 6.1 Sequences . . . . . . . . . . 6.2 Series . . . . . . . . . . . . 6.3 The Integral Test . . . . . . 6.4 Alternating Series . . . . . . 6.5 Comparison Tests . . . . . . 6.6 Absolute Convergence . . . 6.7 The Ratio and Root Tests . . 6.8 Power Series . . . . . . . . 6.9 Calculus with Power Series . 6.10 Taylor Series . . . . . . . . 6.11 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 123 128 130 132 . . . . . . . . . . . 137 138 144 148 153 155 157 159 161 164 166 169 Additional Material 175 Table of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Selected Exercise Answers 183 Index 211 Introduction and Review 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 quickly and effectively 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. 1 Unit 1: Inverse Functions 1.1 Inverse Functions In mathematics, an inverse is a function that serves to “undo” another function. That is, if f (x) produces y, then putting y into the inverse of f produces the output x. A function f that has an inverse is called invertible and the inverse is denoted by f −1 . It is best to illustrate inverses using an arrow diagram: →a 2 →b 3 →c 4 →d →1 b →2 c →3 d →4 a 1 Notice how f maps 1 to a, and f −1 undoes this, that is, f −1 maps a back to 1. Don’t confuse f −1 (x) with 1 . exponentiation: the inverse f −1 is different from f (x) Not every function has an inverse. It is easy to see that if a function f (x) is going to have an inverse, then f (x) never takes on the same value twice. We give this property a special name. A function f (x) is called one-to-one if every element of the range corresponds to exactly one element of the domain. Similar to the Vertical Line Test (VLT) for functions, we have the Horizontal Line Test (HLT) for the one-to-one property. Theorem 1.1: The Horizontal Line Test A function is one-to-one if and only if there is no horizontal line that intersects its graph more than once. Example 1.2: Parabola is Not One-to-one The parabola f (x) = x2 it not one-to-one because it does not satisfy the horizontal line test. For example, the horizontal line y = 1 intersects the parabola at two points, when x = −1 and x = 1. We now formally define the inverse of a function. 3 4 Unit 1: Inverse Functions Definition 1.3: Inverse of a Function Let f (x) and g(x) be two one-to-one functions. If ( f ◦ g)(x) = x and (g ◦ f )(x) = x then we say that f (x) and g(x) are inverses of each other. We denote g(x) (the inverse of f (x)) by g(x) = f −1 (x). Thus, if f maps x to y, then f −1 maps y back to x. This gives rise to the cancellation formulas: f −1 ( f (x)) = x, f ( f −1 (x)) = x, for every x in the domain of f (x), for every x in the domain of f −1 (x). Example 1.4: Finding the Inverse at Specific Values If f (x) = x9 + 2x7 + x + 1, find f −1 (5) and f −1 (1). Solution. Rather than trying to compute a formula for f −1 and then computing f −1 (5), we can simply find a number c such that f evaluated at c gives 5. Note that subbing in some simple values (x = −3, −2, 1, 0, 1, 2, 3) and evaluating f (x) we eventually find that f (1) = 19 +2(17 ) +1 +1 = 5 and f (0) = 1. Therefore, f −1 (5) = 1 and f −1 (1) = 0. ♣ To compute the equation of the inverse of a function we use the following guidelines. Guidelines for Computing Inverses 1. Write down y = f (x). 2. Solve for x in terms of y. 3. Switch the x’s and y’s. 4. The result is y = f −1 (x). Example 1.5: Finding the Inverse Function We find the inverse of the function f (x) = 2x3 + 1. Solution. Starting with y = 2x3 + 1 we solve for x as follows: 3 y − 1 = 2x Therefore, f −1 (x) = r 3 x−1 . 2 → y−1 = x3 2 → x= r 3 y−1 . 2 ♣ This example shows how to find the inverse of a function algebraically. But what about finding the inverse of a function graphically? Step 3 (switching x and y) gives us a good graphical technique to find the inverse, namely, for each point (a, b) where f (a) = b, sketch the point (b, a) for the inverse. More formally, to obtain f −1 (x) reflect the graph f (x) about the line y = x. 1.2. Derivatives of Inverse Functions 5 y x Exercises for 1.1 Exercise 1.1.1 Is the function f (x) = |x| one-to-one? Exercise 1.1.2 Find a formula for the inverse of the function f (x) = x+2 . x−2 1.2 Derivatives of Inverse Functions Suppose we wanted to find the derivative of the inverse, but do not have an actual formula for the inverse function? Then we can use the following derivative formula for the inverse evaluated at a. Derivative of f −1 (a) Given an invertible function f (x), the derivative of its inverse function f −1 (x) evaluated at x = a is:  ′ f 1 (a) = 1 f ′ [ f −1 (a)] To see why this is true, start with the function y = f −1 (x). Write this as x = f (y) and differentiate both sides implicitly with respect to x using the chain rule: 1 = f ′ (y) · dy . dx Thus, dy 1 = ′ , dx f (y) 6 Unit 1: Inverse Functions but y = f −1 (x), thus, At the point x = a this becomes: 1  ′ f −1 (x) = f ′ [ f −1 (x)]  ′ f −1 (a) = f ′ [ f −1 (a)] . 1 Example 1.6: Derivatives of Inverse Functions  ′ Suppose f (x) = x5 + 2x3 + 7x + 1. Find f −1 (1). Solution. First we should show that f −1 exists (i.e. that f is one-to-one). In this case the derivative f ′ (x) = 5x4 + 6x2 + 7 is strictly greater than 0 for all x, so f is strictly increasing and thus one-to-one. It’s difficult to find the inverse of f (x) (and then take the derivative). Thus, we use the above formula evaluated at 1:  −1 ′ 1 f (1) = ′ −1 . f [ f (1)] Note that to use this formula we need to know what f −1 (1) is, and the derivative f ′ (x). To find f −1 (1) we make a table of values (plugging in x = −3, −2, −1, 0, 1, 2, 3 into f (x)) and see what value of x gives 1. We omit the table and simply observe that f (0) = 1. Thus, f −1 (1) = 0. Now we have: And so, f ′ (0) = 7. Therefore,  ′ f −1 (1) =  1 f ′ (0) . ′ 1 f −1 (1) = . 7 ♣ Exercises for 1.2 Exercise 1.2.1 Suppose f (x) = x3 + 4x + 2. Find the slope of the tangent line to the graph of g(x) = x f −1 (x) at the point where x = 7. In the following, verify that the given functions are inverses. Exercise 1.2.2 f (x) = 2x + 6 and g(x) = 21 x − 3 Exercise 1.2.3 f (x) = x2 + 6x + 11, x ≥ 3 and √ g(x) = x − 2 − 3, x ≥ 2 1.2. Derivatives of Inverse Functions Exercise 1.2.4 f (x) = g(x) = 7 3 , x 6= 5 and x−5 3 + 5x , x 6= 0 x Exercise 1.2.5 f (x) = x+1 , x 6= 1 and g(x) = f (x) x−1 In the following, an invertible function f (x) is given along with a point that lies on its graph. Evaluate at the indicated value. ′ f −1 (x) Exercise 1.2.6 f (x) = 5x + 10 Point= (2, 20) ′ Evaluate f −1 (20) Exercise 1.2.9 f (x) = x3 − 6x2 + 15x − 2 Point= (1, 8) ′ Evaluate f −1 (8) Exercise 1.2.7 f (x) = x2 − 2x + 4, x ≥ 1 Point= (3, 7) ′ Evaluate f −1 (7) Exercise 1.2.10 f (x) = Exercise 1.2.8 f (x) = sin 2x, −π /4 ≤ x ≤ π /4 √ Point= (π /6, 3/2) ′ √ Evaluate f −1 ( 3/2) 1 ,x≥0 1 + x2 Point= (1, 1/2) ′ Evaluate f −1 (1/2) Exercise 1.2.11 f (x) = 6e3x Point= (0, 6) ′ Evaluate f −1 (6) In the following, compute the derivative of the given function. Exercise 1.2.12 h(t) = sin−1 (2t) Exercise 1.2.13 f (t) = sec−1 (2t) Exercise 1.2.14 g(x) = tan−1 (2x) Exercise 1.2.15 f (x) = x sin−1 x Exercise 1.2.16 g(t) = sint cos−1 t Exercise 1.2.17 f (t) = lntet Exercise 1.2.18 h(x) = sin−1 x cos−1 x √ Exercise 1.2.19 g(x) = tan−1 ( x) Exercise 1.2.20 f (x) = sec−1 (1/x) Exercise 1.2.21 f (x) = sin(sin−1 x) Exercises 1.2.2 to 1.2.21 were adapted by Lyryx from APEX Calculus, Version 3.0, written by G. Hartman. This material is released under Creative Commons license CC BY-NC ( ). See the Copyright and Revision History pages in the front of this text for more information. 8 Unit 1: Inverse Functions 1.3 Exponential Functions An exponential function is a function of the form f (x) = ax , where a is a constant. Examples are 2x , 10x and (1/2)x. To more formally define the exponential function we look at various kinds of input values. It is obvious that a5 = a · a · a · a · a and a3 = a · a · a, but when we consider an exponential function ax we can’t be limited to substituting integers for x. What does a2.5 or a−1.3 or aπ mean? And is it really true that a2.5 a−1.3 = a2.5−1.3 ? The answer to the first question is actually quite difficult, so we will evade it; the answer to the second question is “yes.” We’ll evade the full answer to the hard question, but we have to know something about exponential functions. You need first to understand that since it’s not “obvious” what 2x should mean, we are really free to make it mean whatever we want, so long as we keep the behavior that is obvious, namely, when x is a positive integer. What else do we want to be true about 2x ? We want the properties of the previous two paragraphs to be true for all exponents: 2x 2y = 2x+y and (2x )y = 2xy . After the positive integers, the next easiest number to understand is 0: 20 = 1. You have presumably learned this fact in the past; why is it true? It is true precisely because we want 2a 2b = 2a+b to be true about the function 2x ....
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