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Differential Calculus 11 CHAPTER M11_HUNG1078_11_AIE_C11.indd Page 569 31/10/13 10:40 AM f-w-147 /203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2_S/203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2 ... Not for Sale Differential Calculus 11 CHAPTER CHAPTER OUTLINE 11.1 Limits How fast is the number of cell phone subscriptions growing? At what rate is the number of 11.2 One-Sided Limits and Limits Involving Internet users increasing? How are home prices changing? These questions and many Infi nity others in the fi elds of business, fi nance, health, political science, psychology, sociology, 11.3 Rates of Change 11.4 Tangent Lines and Derivatives and economics can be answered by using calculus. See Exercise 65 on page 630, 11.5 Techniques for Finding Derivatives Example 12 on page 658 , and Exercise 72 on page 660 . 11.6 Derivatives of Products and Quotients 11.7 The Chain Rule 11.8 Derivatives of Exponential and Logarithmic Functions 11.9 Continuity and Differentiability CASE STUDY 11 Price Elasticity of Demand The algebraic problems considered in earlier chapters dealt with static situations: What is the revenue when x items are sold? How much interest is earned in 2 years? What is the equilibrium price? Calculus, on the other hand, deals with dynamic situations: At what rate is the economy growing? How fast is a rocket going at any instant after liftoff? How quickly can production be increased without adversely affecting profi ts? The techniques of calculus will allow us to answer many questions like these that deal with rates of change. 569 Copyright Pearson. All rights reserved. M11_HUNG1078_11_AIE_C11.indd Page 570 31/10/13 10:40 AM f-w-147 /203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2_S/203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2 ... Not for Sale 570 CHAPTER 11 Differential Calculus The key idea underlying the development of calculus is the concept of limit, so we begin by studying limits. 11.1 Limits We have often dealt with a problem like this: “Find the value of the function f (x) when x = a. ” The underlying idea of “limit,” however, is to examine what the function does near x = a, rather than what it does at x = a. If you would like to refresh your under- standing of functions and functional notation, see Chapter 3 . Example 1 The function 2x2 - 3x - 2 f (x) = x - 2 is not defi ned when x = 2. (Why?) What happens to the values of f (x) when x is very close to 2? Solution Evaluate f at several numbers that are very close to x = 2, as in the following table: x 1.99 1.999 2 2.0001 2.001 f (x) 4.98 4.998 — 5.0002 5.002 The table suggests that as x gets closer and closer to 2 from either direction, the corresponding value of f (x) gets closer and closer to 5. In fact, by experimenting further, you can convince yourself that the values of f (x) can be made as close as you want to 5 by taking values of x close enough to 2. This situation is usually described by saying “The limit of f (x) as x approaches 2 is the number 5,” which is written symbolically as 2x2 - 3x - 2 Checkpoint 1 lim f (x) = 5, or equivalently, lim = 5. S S - 1 x 2 x 2 x 2 Use a calculator to estimate The graph of f shown in Figure 11.1 also shows that lim f (x) = 5 . x3 + x2 - 2x xS2 lim xS1 x - 1 by completing the following table: f(x) 7 x f (x) 6 .9 5 As x approaches 2, the 4 values of f (x) approach 5. .99 3 .999 2 1.0001 1 x –4 –2 –1–3 1 2 34 1.001 –1 1.01 Figure 11.1 1.1 Answers to Checkpoint exercises are found at the end of the section. The informal defi nition of “limit” that follows is similar to the situation in Example 1 , but now f is any function, and a and L are fi xed real numbers (in Example 1 , a = 2 and L = 5 ). Copyright Pearson. All rights reserved. M11_HUNG1078_11_AIE_C11.indd Page 571 31/10/13 10:40 AM f-w-147 /203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2_S/203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2 ... Not for Sale 11.1 Limits 571 Limit of a Function Let f be a function, and let a and L be real numbers. Assume that f (x) is defi ned for all x near x = a. Suppose that as x takes values very close (but not equal) to a (on both sides of a ), the cor- responding values of f (x) are very close (and possibly equal) to L and that the values of f (x) can be made as close as you want to L for all values of x that are close enough to a . Then the number L is the limit of the function f (x) as x approaches a , which is written lim f (x) = L. xSa This defi nition is informal because the expressions “near,” “very close,” and “as close as you want” have not been precisely defi ned. In particular, the tables used in Example 1 and the next set of examples provide strong intuitive evidence, but not a proof, of what the limits must be. Example 2 If f (x) = x2 + x + 1, what is lim f (x) ? xS3 Solution Make a table showing the values of the function at numbers very close to 3: x approaches 3 from the left S 3 d x approaches 3 from the right x 2.9 2.99 2.9999 3 3.0001 3.01 3.1 f (x) 12.31 12.9301 12.9993 . 13.0007 . 13.0701 13.71 f (x) approaches 13 f (x) approaches 13 The table suggests that as x approaches 3 from either direction, f (x) gets closer and closer to 13 and, hence, that lim f (x) = 13, or equivalently, lim (x2 + x + 1) = 13. xS3 xS3 Note that the function f (x) is defi ned when x = 3 and that f (3) = 32 + 3 + 1 = 13. So in this case, the limit of f (x) as x approaches 3 is f (3), the value of the function at 3. (a) Example 3 Use a graphing calculator to fi nd x - 3 lim - . xS3 ex 3 - 1 Solution There are two ways to estimate the limit. x - 3 Graphical Method Graph f (x) = in a very narrow window near x = 3. Use ex-3 - 1 (b) the trace feature to move along the graph and observe the y -coordinates as x gets very close to 3 from either side. Figure 11.2 suggests that lim f (x) = 1 . Figure 11.2 xS3 Copyright Pearson. All rights reserved. M11_HUNG1078_11_AIE_C11.indd Page 572 31/10/13 10:40 AM f-w-147 /203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2_S/203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2 ... Not for Sale 572 CHAPTER 11 Differential Calculus Numerical Method Use the table feature to make a table of values for f (x) when x is very close to 3. Figure 11.3 shows that when x is very close to 3, f (x) is very close to 1. (The table displays “error” at x = 3 because the function is not defi ned when x = 3.) Thus, it appears that - x 3 = lim - 1. xS3 ex 3 - 1 Figure 11.3 The function has the limit 1 as x approaches 3, even though f (3) is not defi ned. Example 4 Find lim f (x), where f is the function whose rule is xS4 0if x is an integer f (x) = e 1if x is not an integer and whose graph is shown in Figure 11.4 . y 2 1 x –1–2 1 2 3 4 5 Figure 11.4 Solution The defi nition of the limit as x approaches 4 involves only values of x that are close, but not equal, to 4—corresponding to the part of the graph on either side of 4, but not at 4 itself. Now, f (x) = 1 for all these numbers (because the numbers very near 4, such as 3.99995 and 4.00002, are not integers). Thus, for all x very close to 4, the corresponding = = value of f (x) is 1, so limS f (x) 1. However, since 4 is an integer, f (4) 0. Therefore, lim f (x) ≠ f ( 4 ) . x 4 xS4 Examples 1 – 4 illustrate the following facts. Limits and Function Values If the limit of a function f (x) as x approaches a exists, then there are three possibilities: 1. f (a) is not defi ned, but lim f (x) is defi ned. ( Examples 1 and 3 ) xSa 2. f (a) is defi ned and lim f (x) = f (a). ( Example 2 ) xSa 3. f (a) is defi ned, but lim f (x) ≠ f (a). ( Example 4 ) xSa Finding Limits Algebraically As we have seen, tables are very useful for estimating limits. However, it is often more effi cient and accurate to fi nd limits algebraically. We begin with two simple functions. Consider the constant function f (x) = 5. To compute lim f (x), you must ask “When x xSa is very close to a , what is the value of f (x)?” The answer is easy because no matter what Copyright Pearson. All rights reserved. M11_HUNG1078_11_AIE_C11.indd Page 573 31/10/13 10:40 AM f-w-147 /203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2_S/203/AW00121/9780321931078_HUNGERFORD/HUNGERFORD_MATHEMATICS_WITH_APPLICATIONS2 ... Not for Sale 11.1 Limits 573 x is, the value of f (x) is always the number 5. As x gets closer and closer to a , the value of f (x) is always 5.
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