Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 23 pps

5.3 Qualitative Interpretation of the Derivative 201 x f ' x f graph of f sign of f ' 0++ 0 (d) graph of f sign of f ' 0++ –1 1 x f ' x f 0 _ –1 1 –1 1 (e) ◆ As you can see, vertically shifting the graph of f doesn’t change its slope. EXERCISE 5.3 Given the graph of f , sketch the graph of f  . Include a number line indicating both the sign of f  and the direction of the graph of f . –2 – 112 x (a)(b) f x f –11 EXERCISE 5.4 Given the graph of f  , sketch two possible graphs of f . Make one of these graphs pass through the origin. x f ' –22 x f ' –1 (a)(b) 202 CHAPTER 5 The Derivative Function PROBLEMS FOR SECTION 5.3 1. Below is a graph of f  , not f . Although the graph is of f  , the questions we ask are about f . (a) For what values of x in the interval [0, 5] is the graph of f increasing? (b) For what values of x in the interval [0, 5] is the graph of f decreasing? (c) Where on the interval [0, 5] is the value of f(x)the smallest? x f ' 12345 2. Below is a graph of g  , not g. Although the graph is of g  , the questions asked are about g. (a) For what values of x in the interval [−2, 6] is the graph of g increasing? (b) For what values of x in the interval [−2, 6] is the graph of g decreasing? (c) Where on the interval [−2, 6] is the value of g(x) the smallest? g' x 24 6–2 5.5 3. On the following page, match each of the eight graphs (a–h) with the correct graph (A–H) of its slope function, i.e., its derivative. When you have completed your work on this problem, compare your answers with those of one of your classmates. If you disagree about an answer, each of you should discuss your reasoning to see if you can come to a consensus on the answers. 5.3 Qualitative Interpretation of the Derivative 203 yyy xxx (a)(b)(c) yy y xxx (d)(e)(f) (g) y (h) y xx y′ y′ y′ xxx AB C y′ y′ y′ xxx DE F y′ y′ xx GH 4. The graph of g(x) = x 2 − 4 looks just like the graph of f(x)= x 2 shifted vertically downward 4 units. Will f  and g  be the same or different? Explain your reasoning. For Problems 5 through 8, sketch the graph of f  . 5. x f –2 6. f x 2 (–1, 1) (2, –2) –2 204 CHAPTER 5 The Derivative Function 7. x f –2 1 8. x f –11234 1 9. How are the graphs of f  and g  related in each of the following situations? Explain your reasoning. (a) g(x) = f(x)+3 (b) g(x) = f(x+3) (c) g(x) = 3f(x) 10. Writing Assignment: You have formed a study group with a few of your friends. One of the people in your study group has been ill for the past 2 1 2 weeks and is concerned about the upcoming examination. She needs to understand the main ideas of the past few weeks. Your essay should be designed to help her. (a) Explain the relationship between average rate of change and instantaneous rate of change, and between secant lines and tangent lines. Your classmate is unclear why the definition of derivative involves some limit with h going to zero. She wants to know why you can’t just set h = 0 to begin with and be done with it. Why does calculus involve this limit business? (b) She also has one specific question. She is not clear about how you get the graph of f  from the graph of f . Just before she got sick we were doing that. She believes that if the graph of f  is increasing, then the graph of f is also increasing. Her Course Assistant says this is wrong, but she claims that sometimes she gets the right answer using this reasoning. What’s the story? How to write this essay: First, think about your friend’s position. Particularly in part (a) see if you can understand what is confusing her and how to clarify it for her. Outline your answer. Then write your essay. Use words precisely. Try to avoid pronouns. For example, do not say “it” is increasing; be specific—what is increasing? Use words to say precisely what you mean. The real purpose of this essay: These are issues that are important for you to understand. We want you to put together what you have learned in your own words. We also want you to learn to write about mathematics by using words precisely to say exactly what you mean. 5.3 Qualitative Interpretation of the Derivative 205 11. In many problems you have been either looking at graphs of trips (position versus time) and producing velocity graphs or, equivalently, looking at graphs of y = f(x) and producing graphs of f  (x), the slope function. In this problem, we give you a velocity graph and ask you to produce the position function given the starting point of the trip. We’ll denote the starting point of the trip by s(0). For each velocity graph, produce three position graphs, one so that s(0) = 0, one with s(0) = 1, and a third with s(0) =−1. (Disclaimer: Note that these graphs of trips are models; that is, it is physically impossible to change from a velocity of 10 units to a velocity of 0 units without traveling at every speed between 10 and 0. In the real world, these graphs would be continuous. But for our purposes, we will work with simplified models.) – 4 –3 –2 –1 1 2 3 4 –20 –10 10 20 30 t v –2 –1 1 2 3 4 5 6 –20 –10 10 20 30 40 t v 12. Below is the graph of a velocity function. Our benchmark time (t = 0) is, as usual, noon. Sketch a graph of position versus time (use our usual conventions: positive position corresponding to east of our benchmark location, negative position corresponding to west) if (a) at 8:00 a.m. s = 0 (b) at 8:00 a.m. s =−20 – 4 –3 –2 –11234 –30 –20 –10 10 20 t (hours) V (mph) 206 CHAPTER 5 The Derivative Function Exploratory Problems for Chapter 5 Running Again 1. Three runners run along the Palouse Path from joining towns in Washington and Idaho. (We discussed these runners in the Exploratory Problems for Chapter 2.) Alicia, Bertha, and Catrina run for one hour. The graphs below give position, s(t),asa function of time for each of the run- ners. The position function gives the distance run from time t = 0 to t. Their position graphs are labeled A, B, and C, respectively. . Sketch velocity graphs for the three women. Label your graphs clearly. . Relate what you’ve done to the graphs of functions and their derivatives. Can you give labels to the graphs making the con- nections between function and derivative clear? . Identify relationships between the graph of s and the corre- sponding graph of v. What characteristic of s assures you that v is positive or zero? What characteristic of s assures you that v is increasing? A B C s (position) in km t (time) in hrs. 1 12.5 2. In this problem we ask you to conjecture about how alterations to f affect the slope function. Here we suggest you arrive at your conjecture by thinking about the relationship between the new and old graphs. At the end of Chapter 7 we will ask you to check your conjectures using the limit definition of derivative. Let f(x)be differentiable on (−∞, ∞) and let k be a con- stant. (a) How is the slope of f(x)+katx=arelated to the slope of f(x)at x = a? If g(x) = f(x)+k, express g  (x) in terms of f  and k. (b) How is the slope of kf(x)at x = a related to the slope of f(x) at x = a? If g(x) = kf(x),express g  (x) in terms of f  and k. (c) If g(x) = f(x+k),express g  (x) in terms of f  and k. Exploratory Problems for Chapter 5 207 (d) Challenge: If g(x) = f(kx), express g  (x) in terms of f  and k. 3. Let f(x)=1/x. (a) Sketch f(x).For what values of x is f(x)increasing, and for what values of x is f(x)decreasing? (b) Sketch the graph of f  (x). (c) Use the limit definition of derivative to find the derivative of f(x)=1/x. (d) Are your answers to parts (b) and (c) consistent? (e) At which point(s) on the graph of f is the slope of the tangent line equal to 1 4 ? 208 CHAPTER 5 The Derivative Function 5.4 INTERPRETING THE DERIVATIVE: MEANING AND NOTATION Getting Perspective We began this chapter with the problem of calculating the velocity of a falling rock at a single instant. Using average velocity to approximate the instantaneous velocity, we applied the strategy of successive approximations and thus narrowed in on the desired quantity. Using analytic methods and a limiting process, we allowed the second data point to get arbitrarily close to the instant in question and found the instantaneous velocity. Adopting a graphical perspective provided us with a new insight; the problem of finding the instantaneous rate of change of f at x = 2 corresponds to finding the slope of the tangent line to f at x = 2. The fundamental challenge is the same, we have one data point but need two. The function itself gives us information about a second data point, Q, which we can slide along the curve to get arbitrarily close to point P = (a, f(a)).Bytaking the limit of the slopes of the secant lines through P and Q as Q approaches P along the curve, we arrive at the slope of the tangent line. We started with the problem of calculating the instantaneous rate of change at a specific instant (t = 2) and found that the same methods could be applied to a generic time, t,to produce a rate function. From a graphical perspective we produced a slope function that can be evaluated at a specific point in the domain of f to get the slope at that point. In Section 5.2 we distinguished between the derivative evaluated at a point and the derivative function. Meaning and Notation Because the limit definitions of f  (c) and f  (x) have been given the dignity of boxes, it may be tempting to memorize them. But there are many different versions of these definitions; if you try to memorize each new version you may find yourself with an annoying abundance of definitions floating around anchorless in your head. It is simpler to focus on the meaning and help yourself along with a picture. Then your knowledge will be portable and robust. Regardless of the form in which it is written, the derivative is simply the limit of the slope of a secant line, y x ,asx tends toward zero. In other words, the slope of the tangent line is lim x→0 y x . EXERCISE 5.5 Which of the following are equal to f  (1)? You may find it helpful to interpret these quotients as slopes of secant lines, labeling the relevant points on the figure on the next page. a) lim x→0 f(1+x) − f(1) x b) lim x→1 f(x)−f(1) x − 1 c) lim h→0 f(1+h) − f(1) h d) lim q→1 f(1)−f(q) 1 − q e) lim x→0 f (x) − f(1) x 5.4 Interpreting the Derivative: Meaning and Notation 209 x f (1, f(1)) Answers are given at the end of this section. Notation Thinking of f  as lim x→0 y x led Leibniz to introduce the notation dy dx or, equivalently, df dx , for the derivative. These latter two notations are helpful in part because they are suggestive of a rate of change. The notation reminds us of the limit we’re computing. Keep in mind that dy dx is not a fraction; it is simply another way of denoting f  . We read dy dx aloud as “dy dx.” If we want to write f  (2) in Leibniz notation, we write df dx    x=2 and read this as “df d x evaluated at x = 2.” If y = f(x),then the derivative function can be denoted in any of the following ways: f  , f  (x), y  , dy dx , df dx . The derivative evaluated at x = 3 can be denoted by: f  (3), y  (3), dy dx     x=3 , df dx     x=3 . We will sometimes find it useful to use the notation d dx to indicate that a derivative is to be taken. d dx is called “operator notation” for the derivative, since it operates on some function and has no independent meaning. For instance d dx (x 2 + 3x) means “the derivative of x 2 + 3x.” Interpreting the Derivative as an Instantaneous Rate of Change Leibniz’s notation, df dx , is very useful when interpreting the derivative as an instantaneous rate of change. In Example 5.11 we interpret the statement f  (2) = 3 in different contexts. ◆ EXAMPLE 5.11 (a) Bottle Calibration: Suppose H(v)gives height (measured in centimeters) as a function of volume v (measured in liters). Then H  (v), also written dH dv , is the function that gives the instantaneous rate of change of height with respect to volume for any volume in its domain. dH dv = lim v→0 H v ; the units for dH dv are cm/L. 210 CHAPTER 5 The Derivative Function H  (2) = 3 tells us that when the volume in the bottle is 2 L, the instantaneous rate of change of height with respect to volume is 3 cm/L. In plain English, at the instant when there are 2 liters of liquid in the bottle, the height of the liquid is increasing at a rate of 3 centimeters per additional liter. (b) Population Growth: Suppose P(t)gives population (measured in thousands of people) as a function of time t (time measured in years, where t = 0 corresponds to January 1, 1950). Then P  (t), also written dP dt , is the function that gives the instantaneous rate of change of population with respect to time. dP dt = lim t→0 P t ; the units for dP dt are thousands of people/year. P  (2) = 3 tells us that when t = 2, the instantaneous rate of change of population with respect to time is 3000 people/year. In plain English, on January 1, 1952, the population is increasing at a rate of 3000 people per year. (c) Position versus Time: Suppose s(t) gives position (measured in miles) as a function of time t (measured in hours). Then s  (t), also written ds dt , is the function that gives the instantaneous rate of change of position with respect to time. ds dt = lim t→0 s t ; the units for ds dt are miles/hour. s  (2) = 3 tells us that at time t = 2, the instantaneous rate of change of position with respect to time is 3 miles/hour. In plain English, the velocity at time t = 2 is 3 mph. (d) Velocity versus Time: Suppose v(t) gives velocity (measured in miles per hour) as a function of time t (measured in hours). Then v  (t), also written dv dt , is the function that gives the instantaneous rate of change of velocity with respect to time. dv dt = lim t→0 v t ; the units for dv dt are mph/hour. v  (2) = 3 tells us that at time t = 2, the instantaneous rate of change of velocity with respect to time is 3 mph/hour.In plain English, the acceleration at time t = 2 is 3 mph/hr. (e) Production Cost versus Amount Produced: Suppose C(x) gives cost (measured in dollars) as a function of x, the number of pounds of material produced (measured in pounds). Then C  (x), also written dC dx , is the function that gives the instantaneous rate of change of cost with respect to number of pounds produced. dC dx = lim x→0 C x ; the units for dC dx are $/lb. C  (2) = 3 tells us that when 2 pounds are being produced, the cost is increasing at a rate of 3 $/lb. When 2 pounds are being produced, the additional cost for producing an additional pound is approximately $3. Economists refer to C  (x) as the marginal cost. The term “marginal cost” is used by economists to mean the additional cost for producing an additional unit or to be C  (x). 12 ◆ 12 For x small, C x and dC dx are approximately equal, but they are not exactly equal. Economists often tend not to bother with the distinction; on the other hand, mathematicians do distinguish between the two. . Your essay should be designed to help her. (a) Explain the relationship between average rate of change and instantaneous rate of change, and between secant lines and tangent lines. Your classmate. the slope of the tangent line. We started with the problem of calculating the instantaneous rate of change at a specific instant (t = 2) and found that the same methods could be applied to a generic. is increasing? Use words to say precisely what you mean. The real purpose of this essay: These are issues that are important for you to understand. We want you to put together what you have learned