DOCSLIB.ORG

Differential and Integral Calculus, with Examples and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential and Integral Calculus, with Examples and Applications TARY TREATISE DIFFERENTIAL AND INTEGRAL AL C CULUS, W ITH EXA MP LES A ND AP PLICA TIONS. B . N S. GEOEGE A . OSB OR E , , PROFESSOR OF MATH EMATIC S IN TH E M ASSACHU SETTS INSTITUTE O F T ECHNOLO GY . H HE TH CO . PUBL S ERS . A D. C , I BOSTON NEW YORK C HICA GO 1903 C OPYR IGHT 1891 , , B B Y GEORGE A . O S ORNE REFA E P C . T IS k n n as a x -b k co and scien H wor , i te ded te t oo for lleges tific c ba o n m m as m s hools , is sed the ethod Of li its , the ost rigorou s and most intelligible form o f presenti ng the first Th e m o f m h a principles Of the s ubject . ethod li its s also the important advantage o f being a familiar method ; for it is n o w so generally introduc ed in the study o f the more ele m n ar anc ma ma c a u n ma be e t y br hes Of the ti s , th t the st de t y assumed to b e fully c onvers ant with it o n beginning the Differential Calculus . o mu a f n a n in The rules r for l e for di fere ti tio Chapter III . in o n e c m i n m a x -b k in b n differ respe t fro those si il r te t oo s, ei g x in m u n a Of x u b n an unc n e pressed ter s Of i ste d , ei g y f tio T are u c a cab all x n o f x. hey th s dire tly ppli le to e pressio s, withou t the aid o f the u sual theorem c oncernin g a function o f a function . A ac u n c f n a n u n fter q iri g the pro esses of di fere ti tio , the st de t i n a V . n uc n a n a n as a Ch pter is i trod ed to the differe ti l ot tio , c onvenient abbreviation o f the c orresponding expression s by n T n a n h a m an dvan differential c oefficie ts . his ot tio s ifest a tages i n the stu dy Of the Integral Calculus an d in its n applicatio s . I n u n u In Chapter X . a d subseq e t pages I have introd ced for 8 Pa a n a o n n a n — c has c n rti l Differe ti ti the ot tio 7 whi h re e tly 6a: n c ome into such ge eral u s e . The chapters On Maxima and Minima h ave been pl aced a a ca n cu as c n a n a fter the ppli tio s to rves , the o sider tio Of th t s ubject is muc h simplified by representi ng the function by n u M x m M e ak n a Of a c . a a and n ma ma b the ordi te rve i i i y t e , ua a an a mm a a C a if desired, with eq l dv t ge i edi tely fter h pter III X . u k m In a X . n a Ca c u a a n b Ch pter , I tegr l l l s, I h ve t e the pro le o f n n M m n n a o f a an a a as a b fi di g the o e t of I erti pl e re , etter illustration o f double i ntegration than that Of finding the The u n m a c m n n area itself. st de t ore re dily o prehe ds the i de n Of a: an d in ub n a pendent variatio y the do le i tegr l, 2 x dx d an in doc d . ( yfi y, th y A a o f a XII n a Ca cu u are d few p ges Ch pter , I tegr l l l s, evoted to a description o f the Hyperbolic Functions together with f n a and a c m a n ma c o r their dif ere ti ls, o p riso is de with the responding Circular Functions . E A . O B R . G . S O N BO TON 1895. S , NT C ONTE S. DIFFERENTIAL CALCULUS. CHAPTER I . FUNCTIONS. Definitio n and Classificatio n o f Functio ns No tatio n Of Emotio n Exam les s . p CHAPTER II . DI ERENTIA E I IE T FF L CO FF C N . mi In r m n Li t . c e e t - m l Diff r n i l ffi i nt . Ex a es 8 10 . e e t a Co e c e p I CHAPTER I I. E DIFFER NTIA TION. l r i Fun tio ns Exam e —2 Differentiatio n o f A geb a c c . pl s 10 1 Diff renti ti n o f Lo arithmic and Ex o nential Functio e a o g p ns . Examples iffer n i io n Of Tri o no metric Fun ctio ns Exam les D e t at g . p . iff r f n r ri n m ri F n i n Ex D e entiatio n o I ve se T go o et c u ct o s . — amples 32 3 7 21 22 Differentiatio n o f In erse Functio n and Functio n Of a , . v Ex m l 3 -40 Functio n . a p es 7 CHAPTER IV. E I E DI ERENTIATI N SUCC S S V FF O . 23 24 . Definiti n nd N t i n , o a o at o 25 . The h i r ffi i Ex m l nt D ffe ential Co e c ent . a p e s ’ 26 . Leibnitz Th r m Ex m l s eo e . a p e s vi TE T CON N S . CHAPTER V. DI EREN I FF T ALS. A M E. Difi erentials as relate d t o Differential Co efficients Differentiatio n b y Differentials Successi e Differentials Ex am les v . p I CHAPTER V . IMP I IT UN TI NS L C F C O . iff r nti tio n o f Im licit Fun i E - D e e a p ct o ns . xamples 52 54 CHAPTER VII . EXPA SI N OF UN TI S N O F C ON . ri ’ Th o r m Exam l Maclau n s e e . p es ’ Ex m l Taylo r s Th eo re m. a p es ’ Rigo ro us Pro o f Of Taylo r s Theo rem ’ ’ Remainder in Taylo r s and Maclaurin s Th eo rems CHAPTER VIII . I DETERMINATE RMS N FO . u f Fra tio n Limiting Val e o 3. c Evaluatio n o f 3. Examples E m le E aluatio n o f 0 co co 00 . xa s v g, , p rm Exam les Evaluatio n o f Exp o nential Fo s . p CHAPTER IX. N PA RTIAL DIFFERENTIATIO . tial Differential C o effi cients Of First Order . Exam 9 6 . Par 5 , 0 — Hi h er rd r Ex m Differential Co efficients o f O e s . a 61 63 . Partial g — ples 80 82 Fun tio ns Of Se eral Variables To ta l Difi erential Of c v . Examples Ex m le s Co nditio n fo r an Exact Differential . a p n tio ns Differentiatio n Of Implicit Fu c . ’ Taylo r s Th e o rem fo r Several V ariables S ii CONTENT . v CHAPTER X . CHANGE OF VARIABLES IN DIFFERENTIAL E COEFFICI NTS. Ch angi ng fro m so to y Ch anging fro m y to z h an in fro m to z Ex am les C g g ac . p CHAPTER XI . REPRESENTATI OF VA RI US URVES ON O C . 4- 85 Rectan ul ar Co -o rdinates 7 . g — - 86 93 . Po lar Co o rdinates CHAPTER XII . D E R E TAN EN D IR TI F U V T AN RMA . C ON O C . G NO L A SYMPTOTES. - i 4 97 Direct o n o f urve ub tan ent and ubno rmal . 9 . C . S g S Example s 98 Differential o effi i n h A r , C c e t o f t e c 99 E uatio n o f th e Tan ent and No rmal Exam les . q g . p 100- 106 A s m to tes Exam les . y p . p CHAPTER XIII . D E N IR TI OF URVA TURE. P I TS OF I F EX I C O C O N N L ON. 10 —109 Directio n Of ur ur 7 . C vat e 110 . Po ints o f Infi exi o n Ex am l . p es CHAPTER XIV. URVATURE. IR E OF URVATURE EV C C CL C .
Load more

Recommended publications
  • Notes on Calculus II Integral Calculus Miguel A. Lerma
    Notes on Calculus II Integral Calculus Miguel A. Lerma November 22, 2002 Contents Introduction 5 Chapter 1. Integrals 6 1.1. Areas and Distances. The Definite Integral 6 1.2. The Evaluation Theorem 11 1.3. The Fundamental Theorem of Calculus 14 1.4. The Substitution Rule 16 1.5. Integration by Parts 21 1.6. Trigonometric Integrals and Trigonometric Substitutions 26 1.7. Partial Fractions 32 1.8. Integration using Tables and CAS 39 1.9. Numerical Integration 41 1.10. Improper Integrals 46 Chapter 2. Applications of Integration 50 2.1. More about Areas 50 2.2. Volumes 52 2.3. Arc Length, Parametric Curves 57 2.4. Average Value of a Function (Mean Value Theorem) 61 2.5. Applications to Physics and Engineering 63 2.6. Probability 69 Chapter 3. Differential Equations 74 3.1. Differential Equations and Separable Equations 74 3.2. Directional Fields and Euler’s Method 78 3.3. Exponential Growth and Decay 80 Chapter 4. Infinite Sequences and Series 83 4.1. Sequences 83 4.2. Series 88 4.3. The Integral and Comparison Tests 92 4.4. Other Convergence Tests 96 4.5. Power Series 98 4.6. Representation of Functions as Power Series 100 4.7. Taylor and MacLaurin Series 103 3 CONTENTS 4 4.8. Applications of Taylor Polynomials 109 Appendix A. Hyperbolic Functions 113 A.1. Hyperbolic Functions 113 Appendix B. Various Formulas 118 B.1. Summation Formulas 118 Appendix C. Table of Integrals 119 Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus.
    [Show full text]
  • A Brief Tour of Vector Calculus
    A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.
    [Show full text]
  • Differentiation Rules (Differential Calculus)
    Differentiation Rules (Differential Calculus) 1. Notation The derivative of a function f with respect to one independent variable (usually x or t) is a function that will be denoted by D f . Note that f (x) and (D f )(x) are the values of these functions at x. 2. Alternate Notations for (D f )(x) d d f (x) d f 0 (1) For functions f in one variable, x, alternate notations are: Dx f (x), dx f (x), dx , dx (x), f (x), f (x). The “(x)” part might be dropped although technically this changes the meaning: f is the name of a function, dy 0 whereas f (x) is the value of it at x. If y = f (x), then Dxy, dx , y , etc. can be used. If the variable t represents time then Dt f can be written f˙. The differential, “d f ”, and the change in f ,“D f ”, are related to the derivative but have special meanings and are never used to indicate ordinary differentiation. dy 0 Historical note: Newton used y,˙ while Leibniz used dx . About a century later Lagrange introduced y and Arbogast introduced the operator notation D. 3. Domains The domain of D f is always a subset of the domain of f . The conventional domain of f , if f (x) is given by an algebraic expression, is all values of x for which the expression is defined and results in a real number. If f has the conventional domain, then D f usually, but not always, has conventional domain. Exceptions are noted below.
    [Show full text]
  • Visual Differential Calculus
    Proceedings of 2014 Zone 1 Conference of the American Society for Engineering Education (ASEE Zone 1) Visual Differential Calculus Andrew Grossfield, Ph.D., P.E., Life Member, ASEE, IEEE Abstract— This expository paper is intended to provide = (y2 – y1) / (x2 – x1) = = m = tan(α) Equation 1 engineering and technology students with a purely visual and intuitive approach to differential calculus. The plan is that where α is the angle of inclination of the line with the students who see intuitively the benefits of the strategies of horizontal. Since the direction of a straight line is constant at calculus will be encouraged to master the algebraic form changing techniques such as solving, factoring and completing every point, so too will be the angle of inclination, the slope, the square. Differential calculus will be treated as a continuation m, of the line and the difference quotient between any pair of of the study of branches11 of continuous and smooth curves points. In the case of a straight line vertical changes, Δy, are described by equations which was initiated in a pre-calculus or always the same multiple, m, of the corresponding horizontal advanced algebra course. Functions are defined as the single changes, Δx, whether or not the changes are small. valued expressions which describe the branches of the curves. However for curves which are not straight lines, the Derivatives are secondary functions derived from the just mentioned functions in order to obtain the slopes of the lines situation is not as simple. Select two pairs of points at random tangent to the curves.
    [Show full text]
  • MATH 162: Calculus II Differentiation
    MATH 162: Calculus II Framework for Mon., Jan. 29 Review of Differentiation and Integration Differentiation Definition of derivative f 0(x): f(x + h) − f(x) f(y) − f(x) lim or lim . h→0 h y→x y − x Differentiation rules: 1. Sum/Difference rule: If f, g are differentiable at x0, then 0 0 0 (f ± g) (x0) = f (x0) ± g (x0). 2. Product rule: If f, g are differentiable at x0, then 0 0 0 (fg) (x0) = f (x0)g(x0) + f(x0)g (x0). 3. Quotient rule: If f, g are differentiable at x0, and g(x0) 6= 0, then 0 0 0 f f (x0)g(x0) − f(x0)g (x0) (x0) = 2 . g [g(x0)] 4. Chain rule: If g is differentiable at x0, and f is differentiable at g(x0), then 0 0 0 (f ◦ g) (x0) = f (g(x0))g (x0). This rule may also be expressed as dy dy du = . dx x=x0 du u=u(x0) dx x=x0 Implicit differentiation is a consequence of the chain rule. For instance, if y is really dependent upon x (i.e., y = y(x)), and if u = y3, then d du du dy d (y3) = = = (y3)y0(x) = 3y2y0. dx dx dy dx dy Practice: Find d x d √ d , (x2 y), and [y cos(xy)]. dx y dx dx MATH 162—Framework for Mon., Jan. 29 Review of Differentiation and Integration Integration The definite integral • the area problem • Riemann sums • definition Fundamental Theorem of Calculus: R x I: Suppose f is continuous on [a, b].
    [Show full text]
  • Unit – 1 Differential Calculus
    Differential Calculus UNIT 1 DIFFERENTIAL CALCULUS Structure 1.0 Introduction 1.1 Objectives 1.2 Limits and Continuity 1.3 Derivative of a Function 1.4 The Chain Rule 1.5 Differentiation of Parametric Forms 1.6 Answers to Check Your Progress 1.7 Summary 1.0 INTRODUCTION In this Unit, we shall define the concept of limit, continuity and differentiability. 1.1 OBJECTIVES After studying this unit, you should be able to : define limit of a function; define continuity of a function; and define derivative of a function. 1.2 LIMITS AND CONTINUITY We start by defining a function. Let A and B be two non empty sets. A function f from the set A to the set B is a rule that assigns to each element x of A a unique element y of B. We call y the image of x under f and denote it by f(x). The domain of f is the set A, and the co−domain of f is the set B. The range of f consists of all images of elements in A. We shall only work with functions whose domains and co–domains are subsets of real numbers. Given functions f and g, their sum f + g, difference f – g, product f. g and quotient f / g are defined by (f + g ) (x) = f(x) + g(x) (f – g ) (x) = f(x) – g(x) (f . g ) (x) = f(x) g(x) f f() x and (x ) g g() x 5 Calculus For the functions f + g, f– g, f. g, the domain is defined to be intersections of the domains of f and g, and for f / g the domain is the intersection excluding the points where g(x) = 0.
    [Show full text]
  • Calculus of Variations
    Calculus of Variations The biggest step from derivatives with one variable to derivatives with many variables is from one to two. After that, going from two to three was just more algebra and more complicated pictures. Now the step will be from a finite number of variables to an infinite number. That will require a new set of tools, yet in many ways the techniques are not very different from those you know. If you've never read chapter 19 of volume II of the Feynman Lectures in Physics, now would be a good time. It's a classic introduction to the area. For a deeper look at the subject, pick up MacCluer's book referred to in the Bibliography at the beginning of this book. 16.1 Examples What line provides the shortest distance between two points? A straight line of course, no surprise there. But not so fast, with a few twists on the question the result won't be nearly as obvious. How do I measure the length of a curved (or even straight) line? Typically with a ruler. For the curved line I have to do successive approximations, breaking the curve into small pieces and adding the finite number of lengths, eventually taking a limit to express the answer as an integral. Even with a straight line I will do the same thing if my ruler isn't long enough. Put this in terms of how you do the measurement: Go to a local store and purchase a ruler. It's made out of some real material, say brass.
    [Show full text]
  • The Calculus of Newton and Leibniz CURRENT READING: Katz §16 (Pp
    Worksheet 13 TITLE The Calculus of Newton and Leibniz CURRENT READING: Katz §16 (pp. 543-575) Boyer & Merzbach (pp 358-372, 382-389) SUMMARY We will consider the life and work of the co-inventors of the Calculus: Isaac Newton and Gottfried Leibniz. NEXT: Who Invented Calculus? Newton v. Leibniz: Math’s Most Famous Prioritätsstreit Isaac Newton (1642-1727) was born on Christmas Day, 1642 some 100 miles north of London in the town of Woolsthorpe. In 1661 he started to attend Trinity College, Cambridge and assisted Isaac Barrow in the editing of the latter’s Geometrical Lectures. For parts of 1665 and 1666, the college was closed due to The Plague and Newton went home and studied by himself, leading to what some observers have called “the most productive period of mathematical discover ever reported” (Boyer, A History of Mathematics, 430). During this period Newton made four monumental discoveries 1. the binomial theorem 2. the calculus 3. the law of gravitation 4. the nature of colors Newton is widely regarded as the most influential scientist of all-time even ahead of Albert Einstein, who demonstrated the flaws in Newtonian mechanics two centuries later. Newton’s epic work Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) more often known as the Principia was first published in 1687 and revolutionalized science itself. Infinite Series One of the key tools Newton used for many of his mathematical discoveries were power series. He was able to show that he could expand the expression (a bx) n for any n He was able to check his intuition that this result must be true by showing that the power series for (1+x)-1 could be used to compute a power series for log(1+x) which he then used to calculate the logarithms of several numbers close to 1 to over 50 decimal paces.
    [Show full text]
  • Elements of Differential Calculus and Optimization
    Elements of differential calculus and optimization. Joan Alexis Glaun`es October 24, 2019 1/29 Differential Calculus in Rn partial derivatives Partial derivatives of a real-valued function defined on Rn : f : Rn ! R. 2 I example : f : R ! R, ( @f (x1; x2) = 4(x1 − 1) + x2 f (x ; x ) = 2(x −1)2+x x +x2 ) @x1 1 2 1 1 2 2 @f (x ; x ) = x + 2x @x2 1 2 1 2 n I example : f : R ! R, 2 2 2 f (x) = f (x1;:::; xn) = (x2 − x1) + (x3 − x2) + ··· + (xn − xn−1) 8 @f (x) = 2(x1 − x2) > @x1 > @f > (x) = 2(x2 − x1) + 2(x2 − x3) > @x2 > @f < (x) = 2(x3 − x2) + 2(x3 − x4) ) @x3 > ··· > @f > (x) = 2(xn−1 − xn−2) + 2(xn−1 − xn) > @xn−1 :> @f (x) = 2(x − x ) @xn n n−1 2/29 Differential Calculus in Rn Directional derivatives n I Let x; h 2 R . We can look at the derivative of f at x in the direction h. It is defined as 0 f (x + "h) − f (x) fh(x) := lim ; "!0 " 0 i.e. fh(x) = g (0) where g(") = f (x + "h) (the restriction of f along the line passing through x with direction h. I The partial derivatives are in fact the directional derivatives in the directions of the canonical basis ei = (0;:::; 1; 0;:::; 0) : @f = f 0 (x): ei @xi 3/29 Differential Calculus in Rn Differential form and Jacobian matrix 0 I The application that maps any direction h to fh(x) is a linear map from Rn to R.
    [Show full text]
  • Differential Calculus of Multivariable Functions
    Math 217 - Calculus 3 for Chemical Engineering Fall 2011 Instructor: Abdullah Hamadeh Part II Differential calculus of multivariable functions 1 Multivariable functions You will be familiar with single variable functions such as those illustrated in Figure 1. z = f (x)= x2 z = f (x)= sinx 1 z = f (x)= x+1 1 x x x 1 − 1 − Figure 1: Functions of a single variable. In each of these cases, the function f ( ) acts as a ‘black box’, operating on a single variable, x, and outputting one, and only one, number, z = f (x), that depends· only on x, as shown in Figure 2 (left). The values of x on which f (x) is defined is termed the domain of f (x): in the three examples above, the domain of f (x) is, respectively, the real line, the real line and the real line excluding the point 1. The range (or image) of f (x) is the set of points to which the function maps its domain: for the three examples− above, the range is, respectively, the set of non-negative real numbers, the segment of the real line [ 1,1] and the real line excluding the point 0. − In this part of the course we shall be analyzing functions of multiple variables, of the form shown in Figure 2 (right). In this example, the function f (x,y) takes two inputs, x and y and outputs one, and only one, number, z = f (x,y). Whereas the domains of the functions f (x) in Figure 1 were subsets of the real line making up the x-axis, the domain of the two variable function z = f (x,y) is composed of subsets of the (two-dimensional) xy- plane, as illustrated in Figure 3.
    [Show full text]
  • The Birth of Calculus: Towards a More Leibnizian View
    The Birth of Calculus: Towards a More Leibnizian View Nicholas Kollerstrom [email protected] We re-evaluate the great Leibniz-Newton calculus debate, exactly three hundred years after it culminated, in 1712. We reflect upon the concept of invention, and to what extent there were indeed two independent inventors of this new mathematical method. We are to a considerable extent agreeing with the mathematics historians Tom Whiteside in the 20th century and Augustus de Morgan in the 19th. By way of introduction we recall two apposite quotations: “After two and a half centuries the Newton-Leibniz disputes continue to inflame the passions. Only the very learned (or the very foolish) dare to enter this great killing- ground of the history of ideas” from Stephen Shapin1 and “When de l’Hôpital, in 1696, published at Paris a treatise so systematic, and so much resembling one of modern times, that it might be used even now, he could find nothing English to quote, except a slight treatise of Craig on quadratures, published in 1693” from Augustus de Morgan 2. Introduction The birth of calculus was experienced as a gradual transition from geometrical to algebraic modes of reasoning, sealing the victory of algebra over geometry around the dawn of the 18 th century. ‘Quadrature’ or the integral calculus had developed first: Kepler had computed how much wine was laid down in his wine-cellar by determining the volume of a wine-barrel, in 1615, 1 which marks a kind of beginning for that calculus. The newly-developing realm of infinitesimal problems was pursued simultaneously in France, Italy and England.
    [Show full text]
  • A Collection of Problems in Differential Calculus
    A Collection of Problems in Differential Calculus Problems Given At the Math 151 - Calculus I and Math 150 - Calculus I With Review Final Examinations Department of Mathematics, Simon Fraser University 2000 - 2010 Veselin Jungic · Petra Menz · Randall Pyke Department Of Mathematics Simon Fraser University c Draft date December 6, 2011 To my sons, my best teachers. - Veselin Jungic Contents Contents i Preface 1 Recommendations for Success in Mathematics 3 1 Limits and Continuity 11 1.1 Introduction . 11 1.2 Limits . 13 1.3 Continuity . 17 1.4 Miscellaneous . 18 2 Differentiation Rules 19 2.1 Introduction . 19 2.2 Derivatives . 20 2.3 Related Rates . 25 2.4 Tangent Lines and Implicit Differentiation . 28 3 Applications of Differentiation 31 3.1 Introduction . 31 3.2 Curve Sketching . 34 3.3 Optimization . 45 3.4 Mean Value Theorem . 50 3.5 Differential, Linear Approximation, Newton's Method . 51 i 3.6 Antiderivatives and Differential Equations . 55 3.7 Exponential Growth and Decay . 58 3.8 Miscellaneous . 61 4 Parametric Equations and Polar Coordinates 65 4.1 Introduction . 65 4.2 Parametric Curves . 67 4.3 Polar Coordinates . 73 4.4 Conic Sections . 77 5 True Or False and Multiple Choice Problems 81 6 Answers, Hints, Solutions 93 6.1 Limits . 93 6.2 Continuity . 96 6.3 Miscellaneous . 98 6.4 Derivatives . 98 6.5 Related Rates . 102 6.6 Tangent Lines and Implicit Differentiation . 105 6.7 Curve Sketching . 107 6.8 Optimization . 117 6.9 Mean Value Theorem . 125 6.10 Differential, Linear Approximation, Newton's Method . 126 6.11 Antiderivatives and Differential Equations .
    [Show full text]