© 2011–2017 by Dale Hoffman, Bellevue College. Author web page: http://scidiv.bellevuecollege.edu/dh/ A free, color PDF version is available online at: http://contemporarycalculus.com/ This text is licensed under a Creative Commons Attribution—Share Alike 3.0 United States License. You are free: • to Share—to copy, distribute, display and perform the work • to Remix—to make derivative works under the following conditions: • Attribution: You must attribute the work in the manner specified by the author (but not in any way that suggests that they endorse you or your work) • Share Alike: If you alter, transform or build upon this work, you must distribute the resulting work only under the same, similar or compatible license. This version edited and typeset (using LATEX) by Jeff Eldridge, Edmonds Community College. Report typographical errors to: [email protected] Tenth printing: March 21, 2017 (corrects 3 minor errors) Ninth printing: December 29, 2016 (corrects 9 minor errors, adds 12 problems to 2.2) Eighth printing: August 1, 2016 (corrects 5 minor errors) Seventh printing: April 1, 2016 (corrects 2 minor errors, adds 6 problems to 1.2) Sixth printing: January 1, 2016 (corrects 6 minor errors) Fifth printing: September 14, 2015 (corrects 1 typographical error and 4 images, clarifies one definition) Fourth printing: March 23, 2015 (corrects 14 errors, adds 12 problems to 3.1 and 18 problems to 3.4) Third printing: September 7, 2014 (corrects 7 typographical errors) Second printing: April 1, 2014 (corrects 21 minor errors and improves image quality throughout) First printing of this version: January 1, 2014 ISBN-13: 978-1494842680 ISBN-10: 1494842688 DALEHOFFMAN CONTEMPORARY CALCULUS Contents 0 Welcome to Calculus 1 0.1 A Preview of Calculus.................... 1 0.2 Lines in the Plane....................... 7 0.3 Functions and Their Graphs................. 20 0.4 Combinations of Functions................. 30 0.5 Mathematical Language................... 45 1 Limits and Continuity 53 1.0 Tangent Lines, Velocities, Growth.............. 53 1.1 The Limit of a Function................... 62 1.2 Properties of Limits...................... 73 1.3 Continuous Functions.................... 84 1.4 Definition of Limit...................... 98 2 The Derivative 109 2.0 Introduction to Derivatives................. 109 2.1 The Definition of Derivative................. 117 2.2 Derivatives: Properties and Formulas........... 130 2.3 More Differentiation Patterns................ 144 2.4 The Chain Rule........................ 155 2.5 Applications of the Chain Rule............... 165 2.6 Related Rates......................... 175 2.7 Newton’s Method....................... 187 2.8 Linear Approximation and Differentials.......... 196 2.9 Implicit and Logarithmic Differentiation......... 208 3 Derivatives and Graphs 217 3.1 Finding Maximums and Minimums............ 217 3.2 Mean Value Theorem..................... 229 3.3 The First Derivative and the Shape of f .......... 238 3.4 The Second Derivative and the Shape of f ........ 248 3.5 Applied Maximum and Minimum Problems....... 258 3.6 Asymptotic Behavior of Functions............. 272 3.7 L’Hôpital’s Rule........................ 283 A AnswersA 1 D Derivative FactsA 29 H How to Succeed in CalculusA 31 T Trigonometry FactsA 35 0 Welcome to Calculus Calculus was first developed more than 300 years ago by Sir Isaac Newton and Gottfried Leibniz to help them describe and understand the rules governing the motion of planets and moons. Since then, thousands of other men and women have refined the basic ideas of calculus, developed new techniques to make the calculations easier, and found ways to apply calculus to a wide variety of problems besides planetary motion. They have used calculus to help understand physical, biological, economic and social phenomena and to describe and solve related problems. The discovery, development and application of calculus is a great intellectual achievement—and now you have the opportunity to share in that achievement. You should feel exhilarated. You may also be somewhat concerned (a common reaction among students just beginning to study calculus). You need to be concerned enough to work to master calculus, yet confident enough to keep going when you (at first) don’t understand something. Part of the beauty of calculus is that it relies upon a few very simple ideas. Part of the power of calculus is that these simple ideas can help us understand, describe and solve problems in a variety of fields. This book tries to emphasize both the beauty and the power. In Section 0.1 (Preview) we will look at the main ideas that will continue throughout the book: the problems of finding tangent lines and computing areas. We will also consider a process that underlies both of those problems: the limiting process of approximating a solution and then getting better and better approximations until we finally get an exact solution. Sections 0.2 (Lines), 0.3 (Functions) and 0.4 (Combinations of Functions) contain review material. These sections emphasize concepts and skills you will need in order to succeed in calculus. You should have worked with most of these concepts in previous courses, but the emphasis and use of the material here may be different than in those earlier classes. Section 0.5 (Mathematical Language) discusses a few key mathematical phrases. It considers their use and meaning and some of their equivalent forms. It will be difficult to understand the meaning and subtleties of calculus if you don’t understand how these phrases are used and what they mean. 0.1 A Preview of Calculus Calculus can be viewed as an attempt—a historically successful attempt—to solve two fundamental problems. In this section we begin to examine geometric forms of those two problems and some fairly simple attempts to solve them. At first, the problems themselves may not appear very interesting or useful—and the methods for solving them may seem crude—but these simple problems and methods have led to one of the most beautiful, powerful and useful creations in mathematics: Calculus. 2 welcome to calculus Finding the Slope of a Tangent Line Suppose we have the graph of a function y = f (x) and we want to find an equation of a line tangent to the graph at a particular point P on the graph (see margin). (We will offer a precise definition of “tangent” in Section 1.0; for now, think of the tangent line as a line that touches the curve at P and stays close to the graph of y = f (x) near P.) We know that P is on the tangent line, so if its x-coordinate is x = a, then the y-coordinate of P must be y = f (a): P = (a, f (a)). The only other information we need to find an equation of the tangent line is its slope, mtan, but that is where the difficulty arises. In algebra, we needed two points in order to determine a slope. So far, we only have the point P. Let’s simply pick a second point, call it Q, on the graph of y = f (x). If the x-coordinate of Q is b, then the y-coordinate is f (b): Q = (b, f (b)). So the slope of the line through P and Q is rise f (b) − f (a) m = = PQ run b − a If we drew the graph of y = f (x) on a wall, put nails at the points P and Q, and laid a straightedge on the nails, then the straightedge would have slope mPQ. But the slope mPQ can be very different from the value we want (the slope mtan of the tangent line). The key idea is that when the point Q is close to the point P, then the slope mPQ should be close to the slope we want, mtan. Physically, if we slide the nail at Q along the f (b)− f (a) graph toward the fixed point P, then the slope, mPQ = b−a , of the straightedge gets closer and closer to the slope, mtan, of the tangent line. If the value of b is very close to a, then the point Q is very close to P, and the value of mPQ is very close to the value of mtan. Rather than defacing walls with graphs and nails, we can instead f (b)− f (a) calculate mPQ = b−a and examine the values of mPQ as b gets closer and closer to a. We say that mtan is the limiting value of mPQ as b gets very close to a, and we write: f (b) − f (a) mtan = lim b!a b − a Eventually we will call the slope mtan of the tangent line the deriva- tive of the function f (x) at the point P, and call this part of calculus differential calculus. Chapters 2 and 3 begin the study of differential calculus. The slope of the tangent line to the graph of a function will tell us important information about the function and will allow us to solve problems such as: • The U.S. Postal Service requires that the length plus the girth of a package not exceed 84 inches. What is the largest volume that can be mailed in a rectangular box? 0.1 a preview of calculus 3 • An oil tanker was leaking oil and a 4-inch-thick oil slick had formed. When first measured, the slick had a radius of 200 feet, and the radius was increasing at a rate of 3 feet per hour. At that time, how fast was the oil leaking from the tanker? Derivatives will even help us solve such “traditional” mathematical problems as finding solutions of equations like x2 = 2 + sin(x) and x9 + 5x5 + x3 + 3 = 0. Problems 1. Sketch the lines tangent to the curve shown below 3. The graph below shows the temperature of a cup at x = 1, 2 and 3.