Calculus I: First Edition ] Tunc Geveci Copyright © 2011 by Tunc Geveci All rights reserved No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of University Readers, Inc First published in the United States of America in 2011 by Cognella, a division of University Readers, Inc Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe 15 14 13 12 11 12345 Printed in the United States of America ISBN: 978-1-935551-42-3 Contents Functions, Limits and Continuity 1.1 Powers of x, Sine and Cosine 1.2 Combinations of Functions 1.3 Limits and Continuity: The Concepts 1.4 The Precise De nitions (Optional) 1.5 The Calculation of Limits 1.6 In nite Limits 1.7 Limits at In nity 1.8 The Limit of a Sequence The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 1 16 31 41 47 58 68 79 Derivative The Concept of the Derivative The Derivatives of Powers and Linear Combinations The Derivatives of Sine and Cosine Velocity and Acceleration Local Linear Approximations and the Di erential The Product Rule and the Quotient Rule The Chain Rule Related Rate Problems Newton’s Method Implicit Di erentiation 93 93 107 121 131 138 148 157 167 174 184 Maxima and Minima 3.1 Increasing/decreasing Behavior and Extrema 3.2 The Mean Value Theorem 3.3 Concavity and Extrema 3.4 Sketching the Graph of a Function 3.5 Applications of Maxima and Minima 193 193 205 214 226 233 Special Functions 4.1 Inverse Functions 4.2 The Derivative of an Inverse Function 4.3 The Natural Exponential and Logarithm 4.4 Arbitrary Bases 4.5 Orders of Magnitude 4.6 Exponential Growth and Decay 4.7 Hyperbolic and Inverse Hyperbolic Functions 4.8 L’Hôpital’s Rule 249 249 262 272 285 291 302 318 332 iii CONTENTS iv The 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Integral The Approximation of Area The De nition of the Integral The Fundamental Theorem of Calculus: Part The Fundamental Theorem of Calculus: Part Integration is a Linear Operation The Substitution Rule The Di erential Equation y = f A Precalculus Review A.1 Solutions of Polynomial Equations A.2 The Binomial Theorem A.3 The Number Line A.4 Decimal Approximations A.5 The Coordinate Plane A.6 Special Angles and Trigonometric Identities 347 347 357 371 385 403 415 427 435 435 440 442 451 457 471 B Some Theorems on Limits and Continuity 481 C The Continuity of an Inverse Function 489 D L’Hôpital’s Rule (A Proof ) 491 E The Natural Logarithm as an Integral 493 F Answers to Some Problems 501 G Basic Derivatives and Integrals 535 Preface This is the rst volume of my calculus series, Calculus I, Calculus II and Calculus III This series is designed for the usual three semester calculus sequence that the majority of science and engineering majors in the United States are required to take Some majors may be required to take only the rst two parts of the sequence Calculus I covers the usual topics of the rst semester: Limits, continuity, the derivative, the integral and special functions such exponential functions, logarithms, and inverse trigonometric functions Calculus II covers the material of the second semester: Further techniques and applications of the integral, improper integrals, linear and separable rst-order di erential equations, in nite series, parametrized curves and polar coordinates Calculus III covers topics in multivariable calculus: Vectors, vector-valued functions, directional derivatives, local linear approximations, multiple integrals, line integrals, surface integrals, and the theorems of Green, Gauss and Stokes An important feature of my book is its focus on the fundamental concepts, essential functions and formulas of calculus Students should not lose sight of the basic concepts and tools of calculus by being bombarded with functions and di erentiation or antidi erentiation formulas that are not signi cant I have written the examples and designed the exercises accordingly I believe that "less is more" That approach enables one to demonstrate to the students the beauty and utility of calculus, without cluttering it with ugly expressions Another important feature of my book is the use of visualization as an integral part of the exposition I believe that the most signi cant contribution of technology to the teaching of a basic course such as calculus has been the e ortless production of graphics of good quality Numerical experiments are also helpful in explaining the basic ideas of calculus, and I have included such data Appendix A and the rst two sections of Chapter provide a review of the essential precalculus material that the student should know in order to meet the challenges of calculus The student should be comfortable with the language and notation that are necessary in order to refer to functions unambiguously That is why I have included such material in the beginning of the rst chapter The main goal of that chapter is to introduce the concepts of limits and continuity, and to provide the student with the necessary tools that are helpful in the calculation of limits I nd it practical to introduce the concepts of limits and continuity simultaneously in terms of the understanding of the concepts and the evaluation of limits Thus, my treatment di ers from the usual calculus text I also treat in nite limits and limits at in nity with more care than the usual calculus text, and provide the student with more tools for the evaluation of such limits The limit of a sequence is treated in this chapter since the language of sequences is convenient in the discussion of Newton’s method and the convergence of certain Riemann sums to the integral The usual calculus text postpones the discussion of sequences to the chapter on in nite series Chapter introduces the derivative I deviate from the usual calculus text by discussing local linear approximations and the di erential at an early stage Indeed, the idea of derivative is v PREFACE vi intimately linked to local linear approximations with a certain form of the error, and that is how the concept is generalized to functions of several variables At this point, the early discussion of local linear approximations is helpful in justifying the identi cation of the derivative of a function with its rate of change and the slope of its graph Such discussion is also helpful in providing plausibility arguments for the product rule and the chain rule Chapter discusses the link between the sign of the derivative, the increasing/decreasing behavior of a function, and its local and global extrema Unlike the usual text, I not start by the stating the theorem on the existence of the absolute extrema of a continuous function on a closed and bounded interval and the Mean Value Theorem Thus, my approach is more practical, and does not give the impression to the student that the only time you can talk about absolute extrema is when you have a continuous function on a closed and bounded interval I also discuss the link between the second derivative, the increasing/decreasing behavior of the rst derivative, and the concavity of a graph Chapter introduces special functions such as exponential and logarithmic functions and inverse trigonometric functions The introduction of inverse functions in the usual text is confusing My introduction is more practical and more careful at the same time I postpone the introduction of the exponential and logarithmic functions to this chapter since I nd it impossible to motivate the signi cance of the natural exponential function before introducing the derivative Besides, powers, sine, cosine, and their combinations are adequate for the illustration of the derivative and its applications prior to this chapter I discuss the di erent orders of magnitude of powers, exponential and logarithms independently of L’Hôpital’s rule I nd this approach much more illuminating than a mechanical applications of L’Hôpital’s rule (that is covered in the last section of the chapter) Chapter introduces the integral I introduce the part of the Fundamental Theorem of Calculus which states that Z b F (x) dx = F (b) F (a) a rst, since that enables the student to compute many integrals before the introduction of the idea of a function that is de ned via an integral Thus, the student has a better chance of understanding the meaning of the part of the Fundamental Theorem which says that Z x d f (t)dt = f (x) dx a (provided that f is continuous) Many texts introduce both parts of the Theorem suddenly, and not present them in a way that establishes the link between the derivative and the integral clearly I nd it amusing, but not helpful, when I see a title such as "total change theorem", as if something other than the Fundamental Theorem is involved My own preference is to cover special functions before the integral since that makes it possible to provide a richer collection of examples and problems in the following chapter On the other hand, some people prefer the elegance of the "late transcendentals" approach whereby the natural logarithm is introduced as an integral The pdf les for the versions of chapter and chapters that introduce the integral before logarithms, exponentials and inverse trigonometric functions will be provided upon request Remarks on some icons: I have indicated the end of a proof by Ơ, the end of an example by Ô and the end of a remark by Supplements: An instructors’ solution manual that contains the solutions of all the problems is available as a PDF le that can be sent to an instructor who has adopted the book The student who purchases the book can access the students’ solutions manual that contains the solutions of odd numbered problems via www.cognella.com PREFACE vii Acknowledgments: Scienti cWorkPlace enabled me to type the text and the mathematical formulas easily in a seamless manner Adobe Acrobat Pro has enabled me to convert the LaTeX les to pdf les Mathematica has enabled me to import high quality graphics to my documents I am grateful to the producers and marketers of such software without which I would not have had he patience to write and rewrite the material in this volume I would also like to acknowledge my gratitude to two wonderful mathematicians who have in uenced me most by demonstrating the beauty of Mathematics and teaching me to write clearly and precisely: Errett Bishop and Stefan Warschawski Last, but not the least, I am grateful to Simla for her encouragement and patience while I spent hours in front a computer screen Tunc Geveci (tgeveci@math.sdsu.edu) San Diego, March 2010 Chapter Functions, Limits and Continuity The rst two sections of this chapter review some basic facts about functions de ned by rational powers of x, polynomials, rational functions and trigonometric functions Appendix A contains additional precalculus review material We will discuss exponential, logarithmic and inverse trigonometric functions in Chapter The main body of the chapter is devoted to the discussion of the fundamental concepts of continuity and limits Roughly speaking, a function f is said to be continuous at a point a if f (x) approximates f (a) when x is close to a It may happen that f (x) approximates a speci c number L if x is close to a but x 6= a, even if f is not de ned at a, or irrespective of the value of f at a The relevant concept is the limit of f at a We will also discuss in nite limits, limits at in nity and the limits of sequences 1.1 Powers of x, Sine and Cosine We will deal with a variety of functions in calculus We will begin by reviewing the relevant terminology and notation Then we will review some basic facts about sin (x), cos (x) and rational powers of x These are the building blocks for a rich collection of functions Terminology and Notation De nition Let D be a subset of the set of real numbers R A real-valued functionof a real variable with domain D is a rule that assigns to each element of D a unique real number We may refer to a function by a letter such as f or g Some functions are special since they occur frequently Such functions have names such as sine or cosine, and their names have speci c abbreviations, such as sin or cos We will denote an arbitrary element of the domain of a function by a letter such as x or t If we choose the letter x, and the function in question is f , then x is the independent variable of f The unique real number that is assigned by f to x is the value of f at x and is denoted by f (x) (read “f of x”) The value of a special function will be denoted by using the abbreviation reserved for that function For example, the value of the sine function at x will be denoted by sin(x) If we set y = f (x), then y is the dependent variable of f We can refer to a function f by the letter that denotes the dependent variable and set y = y (x) (= f (x)) Example Let f (x) = where x 6= x CHAPTER FUNCTIONS, LIMITS AND CONTINUITY We can express the domain of f as ( 4, 0) ^ (0, +4), the union of the interval ( 4, 0) and (0, +4) (Section A3 of Appendix A contains a review of the number line and intervals) The independent variable is x If we set y = 1/x, the dependent variable of f is y We can replace x by any nonzerosreal number to obtain the corresponding value of the function For example, the value of f at is s f ( 2) = s = 0.707107, rounded to signi cant digits (we count the number of signi cant digits of a decimal starting with the rst nonzero digit, as discussed in Section s A4 of Appendix A) Thus, the valuesof the dependent variable y that corresponds to the value of the independent variable is 1/ Ô Example Assume that a car is traveling at a constant speed of 60 miles per hour If we denote time by t (in hours), the distance s covered by the car in t hours is 60t miles Let us set s = f (t) = 60t The letter t denotes the independent variable and can be assigned any nonnegative real number Thus, the domain of f is the set of all nonnegative real numbers and can be expressed as the interval [0, 4) The letter s denotes a dependent variable Ô Example The surface area A of a sphere of radius r is 4 r2 Let us set A = g (r) = 4 r2 for any r > The domain of the function g is the interval (0, +4) The independent variable is r and the dependent variable is A We may choose to refer to the function by the letter that denotes the dependent variable, and set A(r) = 4 r2 Ô Example Let ½ f (x) = x if x < 1, x + if x We have de ned a function whose domain is the set of real numbers R, since the above rule assigns a unique value to any x R, even though the expression for f (x) is not the same for each x R For example, f ( 1) = since < 1, f (1) = + = 2, and f (2) = + = since > We will refer to such a function as a piecewise de ned function Ô Eventually, we will consider relationships between variable entities that need not be real numbers For example, the variable in question can be a point whose coordinates in the Cartesian coordinate plane are real-valued functions of a real variable We will speak of real-valued functions of a real variable simply as functions, until we consider more general relationships Assume that f (x) is a single expression for each x The natural domain of f consists of all x R such that the expression f (x) is a real number We may refer to “the function f (x)” In this case, it should be understood that the domain of the function f is its natural domain For example, if f (x) = 1/x, the natural domain of f consists of all nonzero real numbers We may refer to f as “the function 1/x” The graph of a function is very helpful in visualizing the relationship between the dependent and independent variable If x denotes the independent variable of the function f and y denotes the dependent variable, the graph of f is the subset of the xy-plane that consists of the points (x, y) where x is in the domain of f and y = f (x): {(x, f (x) : x is in the domain of f } Example Let f (x) = 1/x, as in Example In the xy-plane, the graph of f consists of all points (x, y) such that x = and y = 1/x We can express the graph of f as ẵ ả ắ x, : x 6= x APPENDIX F ANSWERS TO SOME PROBLEMS 524 Answers to Some Problems of Section 4.3 (4, +4) b) ¡ ¢ ln e4 = (x ³ ´ 34 ln e34/5 = 23 a) e ln(4) = ´ ³ ln x1/5 = ln (x) 11 2/3 e ¡ ¢ 1) (x + 1)3 x2 + x2 17 19 ln (x 2) x ¢ ¡ ln x + 16 x f (x) = x + 16 21 a) 8x + +4 x+1 x x2 x ¡ ¢4 x + 16 8x 3s 2 + 16 x (x 4) x 1¡ x e e x x2 x 3x2 e x ¢ 29 x3 e x ex (2ex + 1)2 31 (x 1) e (x 33 cos (x) sin (x) e x2 1)2 cos2 (x) 35 x + 2x + x + x2 + e x/2 cos ³x´ e x/2 sin ³x´ Answers to Some Problems of Section 4.4 3 15 x2 = ln (3) x3 x2 17 ¡ ¢ d 2x log10 x2 + = dx (x + 1) ln (10) 7.102/3 22/3 p 11 ± log2 (14) 13 d 1/x 10 = dx d dx 19 ln (10) 1/x 10 x2 = ¶ x 27 ¢ ¡ ln x2 + + ln (x + 1) f (x) = x 2x + x + x2 + + b) 15 f (x) = ln (x + 2) f (x) = x+2 8x + 16 25 13.There is no solution ¶ x d log10 dx x+4 1 (x 1) ln (10) (x + 4) ln (10) ¶ 525 y 21 s d x dx s s = 3x 140 100 d x = x dx x4 gx 80 23 f x xΠ hx x2 40 25 x Answers to Some Problems of Section 4.5 a) y ex/2 x$0+ x ex/2 lim x$+4 x ex/2 = 4, x$0 x ex/2 = +4, lim = x$ x = +4, lim lim e^3 40 80 x -1 The vertical axis is a vertical asymptote for the graph of f a) b) The function is decreasing on ( 4, 0), de2 lim xe x /4 = creasing on (0, 2], and increasing on [2, +4) x$±4 Therefore, f has a local minimum at The x-axis is a horizontal asymptote for the c) graph of f at ±4 s 2], b) The function £is decreasing s s Ô on ( 4, 2, , decreasing on increasing on s [ 2, +4) Thus, f has a local (and global) s minimumsat - 2, and a local (and global) maximum at c) The sgraph of f is concave down on s 6], concave up on [ 6, 0], ( 4, sconcave Ê s Ô down on 0, , concave up on [ 6, +4) Thus,s the x-coordinates of the in ection points s 6, 0, are d) a) y e/2 ln (x) = x$0+ x1/3 lim and x ln (x) = x$+4 x1/3 y lim The vertical axis is a vertical asymptote for the graph of f , and the horizontal axis is a horizontal asymptote at +4 -6^(1/2) b) The function is increasing on (0, e3 ] and decreasing on [e3 , +4) Thus, f has a local (and absolute) maximum at e3 -2^(1/2) 2^(1/2) -1 6^(1/2) x APPENDIX F ANSWERS TO SOME PROBLEMS 526 a) lim f (x) = 10, lim f (x) = x$+4 at (f (1) = e2 ) The function does not have an absolute maximum on (0, +4) x$ y The line y = 10 is a horizontal asymptote for the graph of f at +4 and the x-axis is a horizontal asymptote for the graph of f at b) f is increasing on the entire number line ( 4, +4) c) The graph of f is concave up on ( 4, ln (2)] and concave down on [ln (2) , +4) Thus, the x-coordinate of the point of in ection is ln (2) (the point of in ection is (ln (2) , 5) 11.f attains ¡ ¢its absolute maximum on (0, +4) d) at e2 (f e2 = 2/e) The function does not have an absolute minimum on (0, +4) e^2 y x 10 y 1.0 2/e 0.5 0.0 -4 -2 ln(2) e^2 x 10 20 x -0.5 f attains its absolute minimum on (0, +4) Answers to Some Problems of Section 4.6 b) y (20) = 164 872 million, y (40) = 271 828 million, y (80) = 738 906 million a) Cet/4 b) ±10e(t c) 2)/4 , ±20e(t 2)/4 c) y y 700 600 500 400 20 10 10 20 t 300 200 100 20 a) Ce b) 10e t/5 (t 1)/5 a) Cet/10 b) 2et/10 a) y (t) = 100e0.025t million 40 60 80 t a) y (t) = 40e 0.03t grams b) y (100) = 925 21 = 991 48 grams, y (50) grams, y (150) = 0.444 360 grams c) 527 y 15 Time t is in years a) Y (t) = 4000 (1.04t ) , 40 30 Y (10) = 5920 98, Y (20) = 8764 49, 12973 6, Y (40) Y (30) = = 19204 20 10 50 11 a) y (t) = 100 150 t 100 grams 2t/50 b) b) y (t) = 4000e0.04t Therefore, = 8902 16, = 5967 30, y (20) = 13280.5, y (40) = 19812 y (10) y (30) We see that continuous compounding has led to somewhat higher numbers c) y (100) = 25 grams, y (200) = 25 grams, y (400) = 0.390 625 grams y 20 000 15 000 13 ¢ ¡ a) The doubling time is y (t) = 1000 2t/4 b) 10 000 5000 y (24) = 64 000, y (36) = 512 000, y (48) = 096 × 106 , y (60) = 276 × 107 10 20 30 40 t Answers to Some Problems of Section 4.7 cosh (ln (2)) = 4 (ln (3)) = 5 d cosh (4x) = sinh (4x) dx 17 19 d dx s ¶ = sinh (x) cosh2 (x) d arccosh (3x) = s dx 9x2 9.+4 11.+4 13.0 15.e3 17.1 19.ln (5/2) Answers to Some Problems of Section 5.1 cosh (x) d arcsinh (x/4) = s dx x + 16 Answers to Some Problems of Section 4.8 APPENDIX F ANSWERS TO SOME PROBLEMS 528 X (4k n 16 32 64 1) = 55 k=1 X cos j=1 ³ ´ j = rn 21 875 23 437 24 218 24 609 n mn = 5X f n |rn 25| 125 562 0.781 25 0.390 625 k k=1 b) n 16 32 64 Sn = n$4 n3 lim 0.2 The area of G is (5) (10) = 25 n 5X f n (k 1) k=1 where f (x) = 10 n 16 32 64 25| We see that the left-endpoint sums and rightendpoint sums are not very accurate Each midpoint sum is the exact value of the area 10 ln = |mn 0 0 mn 25 25 25 25 ¶ ¶ n n 0.2 , n |l/n 25| 125 562 0.781 25 0.390 625 ¶ n 5X rn = f k n n k=1 ¶ 2x ln 28 125 26 562 25 781 25 390 1X mn = f n 1+ k k=1 n 16 32 64 ¶ ¯3 ¯ ln (2) ln (5) 2 813 28 × 10 953 17 × 10 882 84 × 10 220 71 × 10 mn 0.235 080 0.235 021 0.235 007 0.235 003 Answers to Some Problems of Section 5.2 1 20 Π Π Π 2 10 2 4 n Π ¶ ¯ mn ¯ 529 Let f (x) = x3 2x2 + n 2X mn = f n 1+ k k=1 n 16 32 64 mn 625 656 25 664 06 666 02 Let f (x) = x2 e ¶ ¶ n ¯ 14 ¯ ¯ mn ¯ 166 67 × 10 041 67 × 10 604 17 × 10 510 42 × 10 n 1X mn = f n 1+ k k=1 n 16 32 64 2 The numbers indicate that such Riemann sums should approximate the integral with desired accuracy, provided that the norm of the partition is small enough x mn 0.486 284 0.486 104 0.486 059 0.486 048 ¶ ¶ n ¯ ¯ ¯5e 10e m (n)¯ 396 94 × 10 988 × 10 496 98 × 10 742 31 × 10 The numbers indicate that such Riemann sums should approximate the integral with desired accuracy, provided that the norm of the partition is small enough Answers to Some Problems of Section 5.3 s 2 1 21/4 35 38 11 31/4 s 13 s s 2+ 15 ln (2) 17 19 21 23 21 Answers to Some Problems of Section 5.4 1 x4 + s + x2 2¶ x cos F has a local p maximum at minimum at 3 /2 11 F 13 p p x2 x2 + + 2x5 x4 + 5 ¶ ¶ F F p /2 and a local ³ ´ ¶ F Answers to Some Problems of Section 5.5 x x + x 120 sin (6t) + cos (2t) 2x4 + 2x2 + 2x3 + 3x 3x2 + 2x + ln (|x|) APPENDIX F ANSWERS TO SOME PROBLEMS 530 15 110 11 s 2 13 y 1 10 1 x 5s 10 s 35 17 s 2+8 Answers to Some Problems of Section 5.6 ¢6 ¡ x +4 12 ¢3/2 2¡ x +1 p x2 11 (x 32 3/2 (x 16) ³x ´ + sin 16) 5/2 + 5 u = sin (x) 5 23 25 ³ 4/3 ´ 28 ³ 4/3 ´ s 3 29 The integral does not exist 27 31 ³ 2/3 13 33 1 + 3 42/3 ´ Ã s ! 3 13 ln (|x + 4|) 15 17 ln2 (x) 35 The picture shows the graph of velocity: s x 2e 19 21 arctan 12 ¶ x 1 arcsin (4x) Π Π 3Π 8 531 The displacement is The distance traveled is 4 Answers to Some Problems of Section 5.7 p x2 + + C 3 cos ³x´ Z t y (t) = + t 10 + e u2 /2 du + 11 Velocity is v (t) = 2x + e 2 Z y (t) = sin (6t) + 10 Position is ¡ ¢ sin u2 du 47 cos (6t) + 10t + 12 12 f (t) = Answers to Some Problems of A1 x = or x = s x = ± 2 s a) x ¡ = ± s2 ¢ ¡ x b) x b) (x 3+ 2i) (x + 2i) a) x = or or b)(x 2) (x 3) (x + 5) s ¢ 11 a) x = or¡ + 2i or ¢ 2i b) (x 2) x2 6x + 13 a) x = ± 2i Answers to Some Problems of A2 ¶ = 20 ¶ (n n = 3 2)(n (4 1)(n) h)3 = 64 (2 + h) h h3 = 12 + 6h + h2 11 (x + h) = x4 + 4x3 h + 6x2 h2 + 4xh3 + h4 48h + 12h2 (x + h) h x3 = 3x2 + 3xh + h2 Answers to Some Problems of A3 â ê a) x R : x < 72 ¢ ¡ b) 4, 72 b) (3, +4) ^ ( 4, 2) â ê a) x R : < x < + 12 ¢ ¡ b) 4, + 12 a) R b) ( 4, +4) a) {x R : x > or x > 2} APPENDIX F ANSWERS TO SOME PROBLEMS 532 a) {x R : < x < 11} b) (5, 11) 11 a) {x R : b) [ 7, 9] x 9} Answers to Some Problems of A4 b) 4.2 × 10 a) 414 214 b) 4.4 × 10 7 a) 22 449 a) 0.166 667 b) 3.3 × 10 b) × 10 a) 5.143 69 a) 0.052632 b) 6.4 × 10 Answers to Some Problems of A5 y y 3,2 3 1 3,3 x 2 1 The point is in the rst quadrant x The point is in the rst quadrant y a) 1, y = + (x 1 x b) y = x 2) 3, c) y The point is in the third quadrant 4,3 2,1 2 y 5,3 1 x The point is in the rst quadrant 11 , y= a) b) y = x c) (x + 4) x 533 c) y 6,2 y x 4, 10 2, 13 a) 23 , y = + (x + 4) 14 b) y = x + 3 c) x 27 a) (4, 6) s b) the parabola intersects the x-axis at + s It ntersects the y-axis at (0, 10) and c) y y 6 4,6 2,6 14 4,2 4 x 10 4 x 20 15 ( 5, 11) 29 17 Both lines have slope 1/2, so that they are a) ( 6, 2) parallel Since they not coincide, they b)The parabola intersects the x s0) saxis at ( 2, not intersect It intersects the y-axis at + and c) 19 b) ( 8/17, 53/17) y c) 0,2 y 6,2 6 0,2 17, 53 17 x 31 a) (x 3)2 + (y + 1)2 = The graph of the equation is a circle of radius centered at The picture is consistent with the claim that (3, 1) the lines are perpendicular 2 x y 21.5 s 23 10 25 a) ( 2, 3) b) The parabola does not intersect the x-axis It intersects the y-axis at (0, 7) 3, x APPENDIX F ANSWERS TO SOME PROBLEMS 534 33 35 (x 2)2 (y + 1)2 (x 4)2 (y 3)2 a) ¶2 + ¶2 = a) = 32 52 5 The graph of the equation is a hyperbola that is centered at (4, 3) The graph of the equation is an ellipse that is b) centered at (2, 1) b) y y 2 x 4,3 2, 2 x Answers to Some Problems of A6 ³ ´ sin = 1 sin 3 ¶ = 1, cos 3 1 1 ¶ 4 = ¶ 4 cos = sin s , 2 s s ¶ 3 2 , cos = , ¶ s 3 1, sec = ¶ ¶ 7 7 = 1, cos = 0, sin 2 ¶ ¶ 7 7 tan and sec are unde ned 2 11 s ¶ ¶ 7 7 = , cos = , sin 6 s s ¶ ¶ 7 7 tan = , sec = 6 ¶ ¶ 3 = sin ¶ 3 tan = s ³ ´ , cos = =0 13 s ¶ ¶ 2 2 = , cos = , sin 3 ¶ ¶ s 2 2 tan = = 3, sec 3 15 cos (4x) = cos4 (x)+sin4 (x) cos2 (x) sin2 (x) 17 1 cos (3x) = cos3 (x) sin2 (x) cos (x) Appendix G Basic Derivatives and Integrals Basic Di erentiation Formulas d r x = rxr dx d sin (x) = cos (x) dx d cos (x) = sin(x) dx d sinh(x) = cosh(x) dx d cosh (x) = sinh(x) dx d tan(x) = dx cos2 (x) d x a = ln (a) ax dx d loga (x) = dx x ln (a) d arcsin (x) = s dx x2 10 d arccos (x) = dx 11 d arctan(x) = dx + x2 s x2 Basic Antidi erentiation Formulas C denotes an arbitrary constant Z Z Z xr dx = xr+1 + C (r 6= r+1 Z cos (x) + C Z Z cos (x) dx = sin (x) + C R R Z dx = ln (|x|) + C x sin (x) dx = 1) 10 sinh(x)dx = cosh(x) + C 535 R cosh(x)dx = sinh(x) + C ex dx = ex + C ax dx = ax + C (a > 0) ln (a) dx = arctan (x) + C + x2 s 1 x2 dx = arcsin (x) + C Index Absolute Value, 6, 445 Acceleration, 129 Antiderivative, 371 Area, 345 area between graphs, 407 area under the graph of a function, 345 signed area, 360 Asymptote horizontal asymptote, 68 oblique asymptote, 74 vertical asymptote, 60 Binomial Theorem, 438 Pascal triangle, 439 Cartesian Coordinates, 455 Circle, 464 unit circle, 465 Completion of the Square, 433 Complex Numbers, 435 Concavity concave down, 212 concave up, 212 in ection point, 215 second derivative test, 216 Continuity, 36, 47 continuity from the left, 38 continuity from the right, 38 continuity on an interval, 38 de nition, 36, 42, 44 jump-discontinuity, 38 proofs of continuity, 479 removable discontinuity, 52 unbounded discontinuity, 40 Cusp, 111 Decimal Approximations, 449 absolute error, 450 chopping, 451 relative error, 452 rounding, 451 signi cant digits, 450 Derivative, 91 as a function, 98 as rate of change, 95 chain rule, 155 constant multiple rule, 112 derivative at a point, 93 higher-order derivatives, 115 power rule, 105 product rule, 146 quotient rule, 148, 150 sum rule, 113 Di erence quotient, 93 Di erential, 140 Di erential Equation, 301, 425 di erential equation y’ = f, 425 general solution, 302, 425 initial-value problem, 302, 426 Ellipse, 465 Error Function erf, 395 Exponential Functions, 270, 283 derivatives of exponential functions, 273, 284 exponentials with arbitrary bases, 496 exponentials with arbitrary basis, 283 natural exponential function, 270, 494 orders of magnitude, 290 Exponential Growth and Decay, 300 compound interest, 308 Euler di erence scheme, 313 population growth, 300 doubling time, 306 Radioactive decay half-life, 307 radioactive decay, 300 Function composition of functions, 23 constant multiple of a function, 18 de nition, domain, even function, graph, identity function, linear combination of functions, 19 536 INDEX 537 linear function, 20 sum rule, 401, 405 natural domain, triangle inequality for integrals, 384 odd function, upper limit, 357 periodic function, 12 Intermediate Value Theorem, 172 polynomial, 19 Intervals, 443 powers of x, 287 closed intervals, 443 product of functions, 16 interior of an interval, 444 quadratic function, 20 open intervals, 443 quotient of functions, 16 Inverse Functions, 247, 487 rational function, 20 derivative of an inverse function, 260 sum of functions, 16 existence and continuity, 487 vertical line test, horizontal line test, 249 Fundamental Theorem of Calculus, 369, 384, inverse cosh, 326 425 inverse cosine, 256, 264 part 1, 369 inverse sine, 254, 262 part 2, 393 inverse sinh, 324 inverse tangent, 258, 266 Graph Sketching, 224 inverse tanh, 328 Hyperbola, 466 Hyperbolic Functions hyperbolic cosine, 316 hyperbolic secant, 323 hyperbolic sine, 316 hyperbolic tangent, 321 Law of Cosines, 475 Leibniz Notation, 101 Limits, 31 calculation of limits, 47, 53 de nition, 33, 42 in nite limits, 58, 71, 87 left-limit, 38 Implicit di erentiation, 182 limit of a sequence, 79 Indeterminate Forms, 64, 66, 74, 291, 294, 296, limits at in nity, 68 299, 330 proofs of limit rules, 479 L’Hôpital’s Rule, 330, 489 right-limit, 38 Inequalities, 441 Linear Approximation, 136 Integral, 355 Lines, 457 additivity w.r.t intervals, 388 perpendicular lines, 460 constant multiple rule, 401, 405 point-slope form of the equation, 458 de nite integral, 373 slope-intercept form of the equation, 457 de nition, 355 Logarithmic Functions inde nite integral, 373 derivatives of logarithmic functions, 278, reverse power rule, 375 286 integrand, 357 logarithmic di erentiation, 281 linearity of the integral, 403, 406 lower limit, 357 logarithms w.r.t arbitrary bases, 285, 497 mean value theorem for integrals, 386 logarthmic growth, 294 Riemann integral, 356 natural logarithm, 276, 491 Riemann sum, 356 Maxima and Minima, 191, 203 left-endpoint sum, 349, 358 absolute maxima and minima, 191, 203 midpoint sum, 350, 358 applications of maxima and minima, 231 right-endpoint sum, 349, 358 applications to economics, 240 signed area, 360 critical point, 196 substitution rule, 413 derivative test, 193 substitution rule for de nite integrals, 419 Fermat’s Theorem, 195 substitution rule for inde nite integrals, local maxima and minima, 191 413 second derivative test, 220, 221 538 stationary point, 195 Mean Value Theorem, 208 Generalized Mean Value Theorem, 489 generalized mean value theorem, 209 Newton, 101 Newton’s Method, 175 Number Line, 442 One-Dimensional Motion, 129 displacement, 380, 429 distance traveled, 381, 410 Parabola, 461 Powers of x, Quadratic Formula, 433 Radian Measure, 10, 469 Related rates, 165 Riemann, 357 Rolle’s Theorem, 207 Secant Line, 92 Sequences, 79 limit of a sequence, 82 Snell’s Law, 238 Special Angles, 469 Speed, 133 Squeeze Theorem, 70, 85 Summation Notation, 345 Tangent Line, 32, 93, 136 Triangle Inequality, 447 Trigonometric Functions cosecant, 22 cosine, 11, 119 cotangent, 22 secant, 21, 152 sine, 11, 119 tangent, 21, 152 trigonometric polynomials, 28, 55 Trigonometric Identities, 473 Addition formulas for sine and cosine, 473 double-angle formulas, 475 Velocity, 129 Vertical Tangent, 110 INDEX ... graphing utility if we wish to emphasize its graphing capabilities, and as a computational utility if we wish to emphasize its computational capabilities If a particular example or exercise requires... concept is the limit of f at a We will also discuss in nite limits, limits at in nity and the limits of sequences 1.1 Powers of x, Sine and Cosine We will deal with a variety of functions in calculus. .. polynomials, rational functions and trigonometric functions Appendix A contains additional precalculus review material We will discuss exponential, logarithmic and inverse trigonometric functions in