Math 111 Calculus I R. Mayer ··· there was far more imagination in the head of Archimedes than in that of Homer. Voltaire[46, page 170] Copyright 2007 by Raymond A. Mayer. Any part of the material protected by this copyright notice may be reproduced in any form for any purpose with- out the permission of the copyright owner. Contents Acknowledgements vii 0 Introduction 1 1 Some Notation for Sets 11 2 Some Area Calculations 19 2.1 The Area Under a Power Function . 19 2.2 Some Summation Formulas . 23 2.3 The Area Under a Parabola . 28 2.4 Finite Geometric Series . 32 1 2.5 Area Under the Curve y = x2 . 36 2.6 ∗Area of a Snowflake. 40 3 Propositions and Functions 51 3.1 Propositions . 51 3.2 Sets Defined by Propositions . 56 3.3 Functions . 58 3.4 Summation Notation . 63 3.5 Mathematical Induction . 65 4 Analytic Geometry 68 4.1 Addition of Points . 68 4.2 Reflections, Rotations and Translations . 73 4.3 The Pythagorean Theorem and Distance. 77 5 Area 83 5.1 Basic Assumptions about Area . 84 5.2 Further Assumptions About Area . 87 iii iv CONTENTS 5.3 Monotonic Functions . 89 5.4 Logarithms. 100 5.5 ∗Brouncker’s Formula For ln(2) . 108 5.6 Computer Calculation of Area . 112 6 Limits of Sequences 116 6.1 Absolute Value . 116 6.2 Approximation . 120 6.3 Convergence of Sequences . 122 6.4 Properties of Limits. 127 6.5 Illustrations of the Basic Limit Properties. 133 6.6 Geometric Series . 141 6.7 Calculation of e . 145 7 Still More Area Calculations 151 7.1 Area Under a Monotonic Function . 151 7.2 Calculation of Area under Power Functions . 154 8 Integrable Functions 160 8.1 Definition of the Integral . 160 8.2 Properties of the Integral . 166 8.3 A Non-integrable Function . 174 8.4 ∗The Ruler Function . 176 8.5 Change of Scale . 179 8.6 Integrals and Area . 183 9 Trigonometric Functions 190 9.1 Properties of Sine and Cosine . 190 9.2 Calculation of π . 200 9.3 Integrals of the Trigonometric Functions . 206 9.4 Indefinite Integrals . 212 10 Definition of the Derivative 219 10.1 Velocity and Tangents . 219 10.2 Limits of Functions . 224 10.3 Definition of the Derivative. 230 CONTENTS v 11 Calculation of Derivatives 237 11.1 Derivatives of Some Special Functions . 237 11.2 Some General Differentiation Theorems. 242 11.3 Composition of Functions . 248 12 Extreme Values of Functions 256 12.1 Continuity . 256 12.2 ∗A Nowhere Differentiable Continuous Function. 259 12.3 Maxima and Minima . 261 12.4 The Mean Value Theorem . 267 13 Applications 272 13.1 Curve Sketching . 272 13.2 Optimization Problems. 277 13.3 Rates of Change . 282 14 The Inverse Function Theorem 287 14.1 The Intermediate Value Property . 287 14.2 Applications . 288 14.3 Inverse Functions . 290 14.4 The Exponential Function . 296 14.5 Inverse Function Theorems . 298 14.6 Some Derivative Calculations . 300 15 The Second Derivative 306 15.1 Higher Order Derivatives . 306 15.2 Acceleration . 310 15.3 Convexity . 313 16 Fundamental Theorem of Calculus 320 17 Antidifferentiation Techniques 328 17.1 The Antidifferentiation Problem . 328 17.2 Basic Formulas . 331 17.3 Integration by Parts . 335 17.4 Integration by Substitution . 340 17.5 Trigonometric Substitution . 345 17.6 Substitution in Integrals . 350 17.7 Rational Functions . 353 vi CONTENTS Bibliography 362 A Hints and Answers 367 B Proofs of Some Area Theorems 372 C Prerequisites 375 C.1 Properties of Real Numbers . 375 C.2 Geometrical Prerequisites . 384 D Some Maple Commands 389 E List of Symbols 393 Index 397 Acknowledgements I would like to thank Joe Buhler, David Perkinson, Jamie Pommersheim, Rao Potluri, Joe Roberts, Jerry Shurman, and Steve Swanson for suggesting nu- merous improvements to earlier drafts of these notes. I would also like to thank Cathy D’Ambrosia for converting my handwritten notes into LATEX. vii Chapter 0 Introduction An Overview of the Course In the first part of these notes we consider the problem of calculating the areas of various plane figures. The technique we use for finding the area of a figure A will be to construct a sequence In of sets contained in A, and a sequence On of sets containing A, such that 1. The areas of In and On are easy to calculate. 2. When n is large then both In and On are in some sense “good approxi- mations” for A. Then by examining the areas of In and On we will determine the area of A. The figure below shows the sorts of sets we might take for In and On in the case where A is the set of points in the first quadrant inside of the circle x2 +y2 = 1. 1 1 1 1 n 1 n 1 n 0 1 0 1 0 1 − 1 In On area(On) area(In)= n 1 2 CHAPTER 0. INTRODUCTION In this example, both of the sets In and On are composed of a finite number 1 of rectangles of width , and from the equation of the circle we can calcu- n late the heights of the rectangles, and hence we can find the areas of In and 1 O . From the third figure we see that area(O )− area(I ) = . Hence if n n n n n = 100000, then either of the numbers area(In) or area(On) will give the area of the quarter-circle with an error of no more than 10−5. This calcula- tion will involve taking many square roots, so you probably would not want to carry it out by hand, but with the help of a computer you could easily find the area of the circle to five decimals accuracy. However no amount of computing power would allow you to get thirty decimals of accuracy from this method in a lifetime, and we will need to develop some theory to get better approximations. In some cases we can find exact areas. For example, we will show that the area of one arch of a sine curve is 2, and the area bounded by the parabola 4 y = x2 and the line y = 1 is . 3 y = 1 A B π 2 4 y = sin(x) area(A)=2 y = x area(B)= 3 However in other cases the areas are not simply expressible in terms of known numbers. In these cases we define certain numbers in terms of areas, for example we will define π = the area of a circle of radius 1, and for all numbers a > 1 we will define ln(a) = the area of the region bounded by the curves y = 0, xy = 1, x = 1, and x = a. 3 We will describe methods for calculating these numbers to any degree of ac- curacy, and then we will√ consider them to be known numbers, just as you probably now think of 2 as being a known number. (Many calculators cal- culate these numbers almost as easily as they calculate square roots.) The numbers ln(a) have many interesting properties which we will discuss, and they have many applications to mathematics and science. Often we consider general classes of figures, in which case we want to find a simple formula giving areas for all of the figures in the class. For example we will express the area of the ellipse bounded by the curve whose equation is x2 y2 + = 1 a2 b2 by means of a simple formula involving a and b. b −a a −b The mathematical tools that we develop for calculating areas, (i.e. the theory of integration) have many applications that seem to have little to do with area. Consider a moving object that is acted upon by a known force F (x) that depends on the position x of the object. (For example, a rocket 4 CHAPTER 0. INTRODUCTION propelled upward from the surface of the moon is acted upon by the moon’s gravitational attraction, which is given by C F (x) = , x2 where x is the distance from the rocket to the center of the moon, and C is some constant that can be calculated in terms of the mass of the rocket and known information.) Then the amount of work needed to move the object from a position x = x0 to a position x = x1 is equal to the area of the region bounded by the lines x = x0, x = x1, y = 0 and y = F (x). y=F(x) R R+H Work is represened by an area In the case of the moon rocket, the work needed to raise the rocket a height H above the surface of the moon is the area bounded by the lines x = R, C x = R + H, y = 0, and y = , where R is the radius of the moon. After we x2 have developed a little bit of machinery, this will be an easy area to calculate. The amount of work here determines the amount of fuel necessary to raise the rocket. Some of the ideas used in the theory of integration are thousands of years old. Quite a few of the technical results in the calculations presented in these notes can be found in the writings of Archimedes(287–212 B.C.), although the way the ideas are presented here is not at all like the way they are presented by Archimedes. In the second part of the notes we study the idea of rate of change.