Applications of Derivatives

Transcription

1 58_Ch04_pp86-60.qd /3/06 :35 PM Page 86 Chapter 4 Applications of Derivatives A n automobile s gas mileage is a function of man variables, including road surface, tire tpe, velocit, fuel octane rating, road angle, and the speed and direction of the wind. If we look onl at velocit s effect on gas mileage, the mileage of a certain car can be approimated b: m(v) v v.8v.7 (where v is velocit) At what speed should ou drive this car to obtain the best gas mileage? The ideas in Section 4. will help ou find the answer. 86

2 Chapter 4 Overview Section 4. Etreme Values of Functions 87 In the past, when virtuall all graphing was done b hand often laboriousl derivatives were the ke tool used to sketch the graph of a function. Now we can graph a function quickl, and usuall correctl, using a grapher. However, confirmation of much of what we see and conclude true from a grapher view must still come from calculus. This chapter shows how to draw conclusions from derivatives about the etreme values of a function and about the general shape of a function s graph. We will also see how a tangent line captures the shape of a curve near the point of tangenc, how to deduce rates of change we cannot measure from rates of change we alread know, and how to find a function when we know onl its first derivative and its value at a single point. The ke to recovering functions from derivatives is the Mean Value Theorem, a theorem whose corollaries provide the gatewa to integral calculus, which we begin in Chapter What ou ll learn about Absolute (Global) Etreme Values Local (Relative) Etreme Values Finding Etreme Values... and wh Finding maimum and minimum values of functions, called optimization, is an important issue in real-world problems. Etreme Values of Functions Absolute (Global) Etreme Values One of the most useful things we can learn from a function s derivative is whether the function assumes an maimum or minimum values on a given interval and where these values are located if it does. Once we know how to find a function s etreme values, we will be able to answer such questions as What is the most effective size for a dose of medicine? and What is the least epensive wa to pipe oil from an offshore well to a refiner down the coast? We will see how to answer questions like these in Section 4.4. DEFINITION Absolute Etreme Values Let f be a function with domain D. Then f c is the (a) absolute maimum value on D if and onl if f f c for all in D. (b) absolute minimum value on D if and onl if f f c for all in D. cos Figure 4. (Eample ) 0 sin Absolute (or global) maimum and minimum values are also called absolute etrema (plural of the Latin etremum). We often omit the term absolute or global and just sa maimum and minimum. Eample shows that etreme values can occur at interior points or endpoints of intervals. EXAMPLE Eploring Etreme Values On p, p, f cos takes on a maimum value of (once) and a minimum value of 0 (twice). The function g sin takes on a maimum value of and a minimum value of (Figure 4.). Now tr Eercise. Functions with the same defining rule can have different etrema, depending on the domain.

3 88 Chapter 4 Applications of Derivatives EXAMPLE Eploring Absolute Etrema The absolute etrema of the following functions on their domains can be seen in Figure 4.. D (, ) Function Rule Domain D Absolute Etrema on D (a) abs min onl (a), (b) 0, No absolute maimum. Absolute minimum of 0 at 0. Absolute maimum of 4 at. Absolute minimum of 0 at 0. (c) 0, Absolute maimum of 4 at. No absolute minimum. D [0, ] (d) 0, No absolute etrema. Now tr Eercise 3. Eample shows that a function ma fail to have a maimum or minimum value. This cannot happen with a continuous function on a finite closed interval. (b) abs ma and min THEOREM The Etreme Value Theorem D (0, ] If f is continuous on a closed interval a, b, then f has both a maimum value and a minimum value on the interval. (Figure 4.3) (, M) (c) abs ma onl D (0, ) a M f() m (, m) Maimum and minimum at interior points b f() M m a b Maimum and minimum at endpoints f() (d) no abs ma or min Figure 4. (Eample ) M f() M m a b Maimum at interior point, minimum at endpoint m a b Minimum at interior point, maimum at endpoint Figure 4.3 Some possibilities for a continuous function s maimum (M) and minimum (m) on a closed interval [a, b].

4 Section 4. Etreme Values of Functions 89 Absolute minimum. No smaller value of f anwhere. Also a local minimum. Local maimum. No greater value of f nearb. Figure 4.4 Classifing etreme values. Local (Relative) Etreme Values a f() c e d Absolute maimum. No greater value of f anwhere. Also a local maimum. Local minimum. No smaller value of f nearb. Figure 4.4 shows a graph with five points where a function has etreme values on its domain a, b. The function s absolute minimum occurs at a even though at e the function s value is smaller than at an other point nearb. The curve rises to the left and falls to the right around c, making f c a maimum locall. The function attains its absolute maimum at d. b Local minimum. No smaller value of f nearb. DEFINITION Local Etreme Values Let c be an interior point of the domain of the function f. Then f c is a (a) local maimum value at c if and onl if f f c for all in some open interval containing c. (b) local minimum value at c if and onl if f f c for all in some open interval containing c. A function f has a local maimum or local minimum at an endpoint c if the appropriate inequalit holds for all in some half-open domain interval containing c. Local etrema are also called relative etrema. An absolute etremum is also a local etremum, because being an etreme value overall makes it an etreme value in its immediate neighborhood. Hence, a list of local etrema will automaticall include absolute etrema if there are an. Finding Etreme Values The interior domain points where the function in Figure 4.4 has local etreme values are points where either f is zero or f does not eist. This is generall the case, as we see from the following theorem. THEOREM Local Etreme Values If a function f has a local maimum value or a local minimum value at an interior point c of its domain, and if f eists at c, then f c 0.

5 90 Chapter 4 Applications of Derivatives Because of Theorem, we usuall need to look at onl a few points to find a function s etrema. These consist of the interior domain points where f 0 or f does not eist (the domain points covered b the theorem) and the domain endpoints (the domain points not covered b the theorem). At all other domain points, f 0 or f 0. The following definition helps us summarize these findings. DEFINITION Critical Point A point in the interior of the domain of a function f at which f 0 or f does not eist is a critical point of f. Thus, in summar, etreme values occur onl at critical points and endpoints. /3 EXAMPLE 3 Finding Absolute Etrema Find the absolute maimum and minimum values of f 3, 3. on the interval [, 3] b [,.5] Figure 4.5 (Eample 3) Solve Graphicall Figure 4.5 suggests that f has an absolute maimum value of about at 3 and an absolute minimum value of 0 at 0. Confirm Analticall We evaluate the function at the critical points and endpoints and take the largest and smallest of the resulting values. The first derivative f has no zeros but is undefined at 0. The values of f at this one critical point and at the endpoints are Critical point value: f 0 0; Endpoint values: f ; f We can see from this list that the function s absolute maimum value is , and occurs at the right endpoint 3. The absolute minimum value is 0, and occurs at the interior point 0. Now tr Eercise. In Eample 4, we investigate the reciprocal of the function whose graph was drawn in Eample 3 of Section. to illustrate grapher failure. EXAMPLE 4 Finding Etreme Values Find the etreme values of f. 4 [ 4, 4] b [, 4] Figure 4.6 The graph of f. 4 (Eample 4) Solve Graphicall Figure 4.6 suggests that f has an absolute minimum of about 0.5 at 0. There also appear to be local maima at and. However, f is not defined at these points and there do not appear to be maima anwhere else. continued

6 Section 4. Etreme Values of Functions 9 Confirm Analticall The function f is defined onl for 4 0, so its domain is the open interval,. The domain has no endpoints, so all the etreme values must occur at critical points. We rewrite the formula for f to find f : f 4. 4 Thus, f 4 3/. 4 3 The onl critical point in the domain, is 0. The value f is therefore the sole candidate for an etreme value. To determine whether is an etreme value of f, we eamine the formula f. 4 As moves awa from 0 on either side, the denominator gets smaller, the values of f increase, and the graph rises. We have a minimum value at 0, and the minimum is absolute. The function has no maima, either local or absolute. This does not violate Theorem (The Etreme Value Theorem) because here f is defined on an open interval. To invoke Theorem s guarantee of etreme points, the interval must be closed. Now tr Eercise 5. While a function s etrema can occur onl at critical points and endpoints, not ever critical point or endpoint signals the presence of an etreme value. Figure 4.7 illustrates this for interior points. Eercise 55 describes a function that fails to assume an etreme value at an endpoint of its domain. 3 /3 0 (a) (b) Figure 4.7 Critical points without etreme values. (a) 3 is 0 at 0, but 3 has no etremum there. (b) 3 3 is undefined at 0, but 3 has no etremum there. EXAMPLE 5 Find the etreme values of Finding Etreme Values 5, f {,. continued

7 9 Chapter 4 Applications of Derivatives [ 5, 5] b [ 5, 0] Figure 4.8 The function in Eample 5. Solve Graphicall The graph in Figure 4.8 suggests that f 0 0 and that f does not eist. There appears to be a local maimum value of 5 at 0 and a local minimum value of 3 at. Confirm Analticall For, the derivative is d { 5 d 4, f d,. d The onl point where f 0 is 0. What happens at? At, the right- and left-hand derivatives are respectivel lim h f h 0 f lim h h 3 lim h 0 h h, h 0 h lim h f h 0 f lim h 5 h 3 h 0 h lim h h 4. h 0 h Since these one-sided derivatives differ, f has no derivative at, and is a second critical point of f. The domain, has no endpoints, so the onl values of f that might be local etrema are those at the critical points: f 0 5 and f 3. From the formula for f, we see that the values of f immediatel to either side of 0 are less than 5, so 5 is a local maimum. Similarl, the values of f immediatel to either side of are greater than 3, so 3 is a local minimum. Now tr Eercise 4. Most graphing calculators have built-in methods to find the coordinates of points where etreme values occur. We must, of course, be sure that we use correct graphs to find these values. The calculus that ou learn in this chapter should make ou feel more confident about working with graphs. Maimum X = Y = [ 4.5, 4.5] b [ 4, ] Figure 4.9 The function in Eample 6. EXAMPLE 6 Using Graphical Methods Find the etreme values of f ln. Solve Graphicall The domain of f is the set of all nonzero real numbers. Figure 4.9 suggests that f is an even function with a maimum value at two points. The coordinates found in this window suggest an etreme value of about 0.69 at approimatel. Because f is even, there is another etreme of the same value at approimatel. The figure also suggests a minimum value at 0, but f is not defined there. Confirm Analticall The derivative f is defined at ever point of the function s domain. The critical points where f 0 are and. The corresponding values of f are both ln ln Now tr Eercise 37.

8 Section 4. Etreme Values of Functions 93 EXPLORATION Finding Etreme Values Let f,.. Determine graphicall the etreme values of f and where the occur. Find f at these values of.. Graph f and f or NDER f,, in the same viewing window. Comment on the relationship between the graphs. 3. Find a formula for f. Quick Review 4. (For help, go to Sections.,., 3.5, and 3.6.) In Eercises 4, find the first derivative of the function.. f 4. f 9 3. g cos ln 4. h e In Eercises 5 8, match the table with a graph of f () f a 0 b 0 c f a does not eist b 0 c a b c f a 0 b 0 c 5 f does not eist does not eist.7 a b c (c) In Eercises 9 and 0, find the limit for 9. lim 3 f In Eercises and, let f lim f 3 3, f {,. a b c (d). Find (a) f, (b) f 3, (c) f.. (a) Find the domain of f. (b) Write a formula for f. a b c (a) a b c (b) Section 4. Eercises In Eercises 4, find the etreme values and where the occur (, ) 3

9 94 Chapter 4 Applications of Derivatives In Eercises 5 0, identif each -value at which an absolute etreme value occurs. Eplain how our answer is consistent with the Etreme Value Theorem h() f() 0 a c c b f() 0 a c b g() 0 a c b In Eercises 8, use analtic methods to find the etreme values of the function on the interval and where the occur.. f ln, g e, 3. h ln, k e, 5. f sin ( p 4 ), 0 7 4p 6. g sec, p 3 p 7. f 5, 3 8. f 3 5, 3 In Eercises 9 30, find the etreme values of the function and where the occur a c b 0 a b 30. h() c g() 0 a c b Group Activit In Eercises 3 34, find the etreme values of the function on the interval and where the occur. 3. f 3, g 5, h 3, 34. k 3, In Eercises 35 4, identif the critical point and determine the local etreme values , 39. {, 40. 3, 0 { 3, , { 6 4, { 4 5, , 43. Writing to Learn The function V 0 6, 0 5, models the volume of a bo. (a) Find the etreme values of V. (b) Interpret an values found in (a) in terms of volume of the bo. 44. Writing to Learn The function P 0 0, 0, models the perimeter of a rectangle of dimensions b 00. (a) Find an etreme values of P. (b) Give an interpretation in terms of perimeter of the rectangle for an values found in (a). Standardized Test Questions You should solve the following problems without using a graphing calculator. 45. True or False If f (c) is a local maimum of a continuous function f on an open interval (a, b), then f (c) 0. Justif our answer. 46. True or False If m is a local minimum and M is a local maimum of a continuous function f on (a, b), then m M. Justif our answer. 47. Multiple Choice Which of the following values is the absolute maimum of the function f () 4 6 on the interval [0, 4]? (A) 0 (B) (C) 4 (D) 6 (E) 0

10 Section 4. Etreme Values of Functions Multiple Choice If f is a continuous, decreasing function on [0, 0] with a critical point at (4, ), which of the following statements must be false? (A) f (0) is an absolute minimum of f on [0, 0]. (B) f (4) is neither a relative maimum nor a relative minimum. (C) f (4) does not eist. (D) f (4) 0 (E) f (4) Multiple Choice Which of the following functions has eactl two local etrema on its domain? (A) f () (B) f () (C) f() (D) f () tan (E) f () ln 50. Multiple Choice If an even function f with domain all real numbers has a local maimum at a, then f ( a) (A) is a local minimum. (B) is a local maimum. (C) is both a local minimum and a local maimum. (D) could be either a local minimum or a local maimum. (E) is neither a local minimum nor a local maimum. Eplorations In Eercises 5 and 5, give reasons for our answers. 5. Writing to Learn Let f 3. (a) Does f eist? (b) Show that the onl local etreme value of f occurs at. (c) Does the result in (b) contradict the Etreme Value Theorem? (d) Repeat parts (a) and (b) for f a 3, replacing b a. 5. Writing to Learn Let f 3 9. (a) Does f 0 eist? (b) Does f 3 eist? (c) Does f 3 eist? (d) Determine all etrema of f. Etending the Ideas 53. Cubic Functions Consider the cubic function f a 3 b c d. (a) Show that f can have 0,, or critical points. Give eamples and graphs to support our argument. (b) How man local etreme values can f have? 54. Proving Theorem Assume that the function f has a local maimum value at the interior point c of its domain and that f (c) eists. (a) Show that there is an open interval containing c such that f f c 0 for all in the open interval. (b) Writing to Learn Now eplain wh we ma sa lim f c c f 0. c (c) Writing to Learn Now eplain wh we ma sa lim f c c f 0. c (d) Writing to Learn Eplain how parts (b) and (c) allow us to conclude f c 0. (e) Writing to Learn Give a similar argument if f has a local minimum value at an interior point. 55. Functions with No Etreme Values at Endpoints (a) Graph the function { sin, 0 f 0, 0. Eplain wh f 0 0 is not a local etreme value of f. (b) Group Activit Construct a function of our own that fails to have an etreme value at a domain endpoint.

11 96 Chapter 4 Applications of Derivatives 4. What ou ll learn about Mean Value Theorem Phsical Interpretation Increasing and Decreasing Functions Other Consequences... and wh The Mean Value Theorem is an important theoretical tool to connect the average and instantaneous rates of change. 0 a f() Slope f'(c) A c Tangent parallel to chord Slope f(b) f(a) b a Figure 4.0 Figure for the Mean Value Theorem. Rolle s Theorem The first version of the Mean Value Theorem was proved b French mathematician Michel Rolle (65 79). His version had f a f b 0 and was proved onl for polnomials, using algebra and geometr. B b Mean Value Theorem Mean Value Theorem The Mean Value Theorem connects the average rate of change of a function over an interval with the instantaneous rate of change of the function at a point within the interval. Its powerful corollaries lie at the heart of some of the most important applications of the calculus. The theorem sas that somewhere between points A and B on a differentiable curve, there is at least one tangent line parallel to chord AB (Figure 4.0). THEOREM 3 Mean Value Theorem for Derivatives If f is continuous at ever point of the closed interval a, b and differentiable at ever point of its interior a, b, then there is at least one point c in a, b at which f c f b f a. b a The hpotheses of Theorem 3 cannot be relaed. If the fail at even one point, the graph ma fail to have a tangent parallel to the chord. For instance, the function f is continuous on, and differentiable at ever point of the interior, ecept 0. The graph has no tangent parallel to chord AB (Figure 4.a). The function g int is differentiable at ever point of, and continuous at ever point of, ecept. Again, the graph has no tangent parallel to chord AB (Figure 4.b). The Mean Value Theorem is an eistence theorem. It tells us the number c eists without telling how to find it. We can sometimes satisf our curiosit about the value of c but the real importance of the theorem lies in the surprising conclusions we can draw from it. A (, ) a, b B (, ) 0 A (, ) a B (, ) int, b f ' (c) = 0 (a) Figure 4. No tangent parallel to chord AB. (b) = f () 0 a c Rolle distrusted calculus and spent most of his life denouncing it. It is ironic that he is known toda onl for an unintended contribution to a field he tried to suppress. b EXAMPLE Eploring the Mean Value Theorem Show that the function f satisfies the hpotheses of the Mean Value Theorem on the interval 0,. Then find a solution c to the equation f c f b f a b a on this interval. continued

12 Section 4. Mean Value Theorem 97 B(, 4) = (, ) A(0, 0) Figure 4. ( Eample ) The function f is continuous on 0, and differentiable on 0,. Since f 0 0 and f 4, the Mean Value Theorem guarantees a point c in the interval 0, for which f c f b f a b a c f f 0 f 0 c. Interpret The tangent line to f at has slope and is parallel to the chord joining A 0, 0 and B, 4 (Figure 4.). Now tr Eercise. EXAMPLE Eploring the Mean Value Theorem Eplain wh each of the following functions fails to satisf the conditions of the Mean Value Theorem on the interval [, ]. (a) f () (b) 3 3 for f { for (a) Note that, so this is just a vertical shift of the absolute value function, which has a nondifferentiable corner at 0. (See Section 3..) The function f is not differentiable on (, ). (b) Since lim f () lim and lim + f () lim +, the function has a discontinuit at. The function f is not continuous on [, ]. If the two functions given had satisfied the necessar conditions, the conclusion of the Mean Value Theorem would have guaranteed the eistence of a number c in (, ) such that f (c) f ( ) f ( ) 0. Such a number c does not eist for the function in ( ) part (a), but one happens to eist for the function in part (b) (Figure 4.3) (a) Figure 4.3 For both functions in Eample, f ( ) f ( ) 0 but neither ( ) function satisfies the conditions of the Mean Value Theorem on the interval [, ]. For the function in Eample (a), there is no number c such that f (c) 0. It happens that f (0) 0 in Eample (b). (b) Now tr Eercise 3.

13 98 Chapter 4 Applications of Derivatives Figure 4.4 ( Eample 3), 0 EXAMPLE 3 Appling the Mean Value Theorem Let f, A, f, and B, f. Find a tangent to f in the interval, that is parallel to the secant AB. The function f (Figure 4.4) is continuous on the interval [, ] and f is defined on the interval,. The function is not differentiable at and, but it does not need to be for the theorem to appl. Since f f 0, the tangent we are looking for is horizontal. We find that f 0 at 0, where the graph has the horizontal tangent. Now tr Eercise 9. Distance (ft) s s f(t) At this point, the car s speed was 30 mph. 5 Time (sec) Figure 4.5 ( Eample 4) (8, 35) t Phsical Interpretation If we think of the difference quotient f b f a b a as the average change in f over a, b and f c as an instantaneous change, then the Mean Value Theorem sas that the instantaneous change at some interior point must equal the average change over the entire interval. EXAMPLE 4 Interpreting the Mean Value Theorem If a car accelerating from zero takes 8 sec to go 35 ft, its average velocit for the 8-sec interval is ft sec, or 30 mph. At some point during the acceleration, the theorem sas, the speedometer must read eactl 30 mph (Figure 4.5). Now tr Eercise. Increasing and Decreasing Functions Our first use of the Mean Value Theorem will be its application to increasing and decreasing functions. Monotonic Functions A function that is alwas increasing on an interval or alwas decreasing on an interval is said to be monotonic there. DEFINITIONS Increasing Function, Decreasing Function Let f be a function defined on an interval I and let and be an two points in I.. f increases on I if f f.. f decreases on I if f f. The Mean Value Theorem allows us to identif eactl where graphs rise and fall. Functions with positive derivatives are increasing functions; functions with negative derivatives are decreasing functions. COROLLARY Increasing and Decreasing Functions Let f be continuous on a, b and differentiable on a, b.. If f 0 at each point of a, b, then f increases on a, b.. If f 0 at each point of a, b, then f decreases on a, b.

14 Section 4. Mean Value Theorem 99 4 Proof Let and be an two points in a, b with. The Mean Value Theorem applied to f on, gives f f f c Function decreasing ' 0 3 Function increasing ' 0 for some c between and. The sign of the right-hand side of this equation is the same as the sign of f c because is positive. Therefore, (a) f f if f 0 on a, b ( f is increasing), or (b) f f if f 0 on a, b ( f is decreasing). 0 ' 0 Figure 4.6 ( Eample 5) EXAMPLE 5 Determining Where Graphs Rise or Fall The function (Figure 4.6) is (a) decreasing on, 0 because 0 on, 0. (b) increasing on 0, because 0 on 0,. Now tr Eercise 5. What s Happening at Zero? Note that 0 appears in both intervals in Eample 5, which is consistent both with the definition and with Corollar. Does this mean that the function is both increasing and decreasing at 0? No! This is because a function can onl be described as increasing or decreasing on an interval with more than one point (see the definition). Saing that is increasing at is not reall proper either, but ou will often see that statement used as a short wa of saing is increasing on an interval containing. [ 5, 5] b [ 5, 5] Figure 4.7 B comparing the graphs of f 3 4 and f 3 4 we can relate the increasing and decreasing behavior of f to the sign of f. (Eample 6) EXAMPLE 6 Determining Where Graphs Rise or Fall Where is the function f 3 4 increasing and where is it decreasing? Solve Graphicall The graph of f in Figure 4.7 suggests that f is increasing from to the -coordinate of the local maimum, decreasing between the two local etrema, and increasing again from the -coordinate of the local minimum to. This information is supported b the superimposed graph of f 3 4. Confirm Analticall The function is increasing where f or 4 3 The function is decreasing where f In interval notation, f is increasing on, 4 3 ], decreasing on 4 3, 4 3, and increasing on 4 3,. Now tr Eercise 7. Other Consequences We know that constant functions have the zero function as their derivative. We can now use the Mean Value Theorem to show conversel that the onl functions with the zero function as derivative are constant functions. COROLLARY Functions with f = 0 are Constant If f 0 at each point of an interval I, then there is a constant C for which f C for all in I.

15 00 Chapter 4 Applications of Derivatives Proof Our plan is to show that f f for an two points and in I. We can assume the points are numbered so that. Since f is differentiable at ever point of,, it is continuous at ever point as well. Thus, f satisfies the hpotheses of the Mean Value Theorem on,. Therefore, there is a point c between and for which f c f f. Because f c 0, it follows that f f. We can use Corollar to show that if two functions have the same derivative, the differ b a constant. COROLLARY 3 Functions with the Same Derivative Differ b a Constant If f g at each point of an interval I, then there is a constant C such that f g C for all in I. Proof Let h f g. Then for each point in I, h f g 0. It follows from Corollar that there is a constant C such that h C for all in I. Thus, h f g C, or f g C. We know that the derivative of f is on the interval,. So, an other function g with derivative on, must have the formula g C for some constant C. EXAMPLE 7 Appling Corollar 3 Find the function f whose derivative is sin and whose graph passes through the point 0,. Since f has the same derivative as g() cos, we know that f () cos C, for some constant C. To identif C, we use the condition that the graph must pass through (0, ). This is equivalent to saing that f(0) cos 0 C C C 3. f cos C The formula for f is f cos 3. Now tr Eercise 35. In Eample 7 we were given a derivative and asked to find a function with that derivative. This tpe of function is so important that it has a name. DEFINITION Antiderivative A function F is an antiderivative of a function f if F f for all in the domain of f. The process of finding an antiderivative is antidifferentiation.

16 Section 4. Mean Value Theorem 0 We know that if f has one antiderivative F then it has infinitel man antiderivatives, each differing from F b a constant. Corollar 3 sas these are all there are. In Eample 7, we found the particular antiderivative of sin whose graph passed through the point 0,. EXAMPLE 8 Finding Velocit and Position Find the velocit and position functions of a bod falling freel from a height of 0 meters under each of the following sets of conditions: (a) The acceleration is 9.8 m sec and the bod falls from rest. (b) The acceleration is 9.8 m sec and the bod is propelled downward with an initial velocit of m sec. (a) Falling from rest. We measure distance fallen in meters and time in seconds, and assume that the bod is released from rest at time t 0. Velocit: We know that the velocit v t is an antiderivative of the constant function 9.8. We also know that g t 9.8t is an antiderivative of 9.8. B Corollar 3, v t 9.8t C for some constant C. Since the bod falls from rest, v 0 0. Thus, C 0 and C 0. The bod s velocit function is v t 9.8t. Position: We know that the position s t is an antiderivative of 9.8t. We also know that h t 4.9t is an antiderivative of 9.8t. B Corollar 3, s t 4.9t C for some constant C. Since s 0 0, C 0 and C 0. The bod s position function is s t 4.9t. (b) Propelled downward. We measure distance fallen in meters and time in seconds, and assume that the bod is propelled downward with velocit of m sec at time t 0. Velocit: The velocit function still has the form 9.8t C, but instead of being zero, the initial velocit (velocit at t 0) is now m sec. Thus, C and C. The bod s velocit function is v t 9.8t. Position: We know that the position s t is an antiderivative of 9.8t. We also know that k t 4.9t t is an antiderivative of 9.8t. B Corollar 3, for some constant C. Since s 0 0, s t 4.9t t C C 0 and C 0. The bod s position function is s t 4.9t t. Now tr Eercise 43.

17 0 Chapter 4 Applications of Derivatives Quick Review 4. (For help, go to Sections.,.3, and 3..) In Eercises and, find eact solutions to the inequalit In Eercises 3 5, let f Find the domain of f. 4. Where is f continuous? 5. Where is f differentiable? In Eercises 6 8, let f. 6. Find the domain of f. 7. Where is f continuous? 8. Where is f differentiable? In Eercises 9 and 0, find C so that the graph of the function f passes through the specified point. 9. f C,, 7 0. g C,, Section 4. Eercises In Eercises 8, (a) state whether or not the function satisfies the hpotheses of the Mean Value Theorem on the given interval, and (b) if it does, find each value of c in the interval (a, b) that satisfies the equation f (c) f (b ) f (a). b a. f () on [0, ]. f () 3 on [0, ] 3. f () 3 on [,] 4. f () on [0, 4] 5. f () sin on [, ] 6. f () ln( ) on [, 4] cos, 0 p 7. f () on [0, p] sin, p p sin, 8. f () on [, 3] /, 3 In Eercises 9 and 0, the interval a b is given. Let A a, f a and B b, f b. Write an equation for (a) the secant line AB. (b) a tangent line to f in the interval a, b that is parallel to AB. 9. f, f, 3. Speeding A trucker handed in a ticket at a toll booth showing that in h she had covered 59 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Wh?. Temperature Change It took 0 sec for the temperature to rise from 0 F to F when a thermometer was taken from a freezer and placed in boiling water. Eplain wh at some moment in that interval the mercur was rising at eactl 0.6 F/sec. 3. Triremes Classical accounts tell us that a 70-oar trireme (ancient Greek or Roman warship) once covered 84 sea miles in 4 h. Eplain wh at some point during this feat the trireme s speed eceeded 7.5 knots (sea miles per hour). 4. Running a Marathon A marathoner ran the 6.-mi New York Cit Marathon in. h. Show that at least twice, the marathoner was running at eactl mph. In Eercises 5, use analtic methods to find (a) the local etrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. 5. f 5 6. g 7. h 8. k 9. f e 0. f e

18 Section 4. Mean Value Theorem 03 In Eercises 3 8, find (a) the local etrema, (b) the intervals on which the function is increasing, and (c) the intervals on which the function is decreasing. 3. f 4 4. g h 6. k f 3 cos 8. g cos 44. Diving (a) With what velocit will ou hit the water if ou step off from a 0-m diving platform? (b) With what velocit will ou hit the water if ou dive off the platform with an upward velocit of m sec? In Eercises 9 34, find all possible functions f with the given derivative. 9. f 30. f 3. f 3 3. f sin 33. f e 34. f, In Eercises 35 38, find the function with the given derivative whose graph passes through the point P. 35. f, 0, P, 36. f, P, 37. f,, P, f cos, P 0, 3 Group Activit In Eercises 39 4, sketch a graph of a differentiable function f that has the given properties. 39. (a) local minimum at,, local maimum at 3, 3 (b) local minima at, and 3, 3 (c) local maima at, and 3, f 3, f 0, and (a) f 0 for, f 0 for. (b) f 0 for, f 0 for. (c) f 0 for. (d) f 0 for. 4. f f 0, f 0 on,, f 0 for, f 0 for. 4. A local minimum value that is greater than one of its local maimum values. 43. Free Fall On the moon, the acceleration due to gravit is.6 m sec. (a) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later? (b) How far below the point of release is the bottom of the crevasse? (c) If instead of being released from rest, the rock is thrown into the crevasse from the same point with a downward velocit of 4 m sec, when will it hit the bottom and how fast will it be going when it does? 45. Writing to Learn The function, 0 f { 0, is zero at 0 and at. Its derivative is equal to at ever point between 0 and, so f is never zero between 0 and, and the graph of f has no tangent parallel to the chord from 0, 0 to, 0. Eplain wh this does not contradict the Mean Value Theorem. 46. Writing to Learn Eplain wh there is a zero of cos between ever two zeros of sin. 47. Unique Solution Assume that f is continuous on a, b and differentiable on a, b. Also assume that f a and f b have opposite signs and f 0 between a and b. Show that f 0 eactl once between a and b. In Eercises 48 and 49, show that the equation has eactl one solution in the interval. [Hint: See Eercise 47.] , 49. ln 0, Parallel Tangents Assume that f and g are differentiable on a, b and that f a g a and f b g b. Show that there is at least one point between a and b where the tangents to the graphs of f and g are parallel or the same line. Illustrate with a sketch. Standardized Test Questions You ma use a graphing calculator to solve the following problems. 5. True or False If f is differentiable and increasing on (a, b), then f (c) 0 for ever c in (a, b). Justif our answer. 5. True or False If f is differentiable and f (c) > 0 for ever c in (a, b), then f is increasing on (a, b). Justif our answer.

19 04 Chapter 4 Applications of Derivatives 53. Multiple Choice If f () cos, then the Mean Value Theorem guarantees that somewhere between 0 and p/3, f () 3 (A) p (B) 3 (C) (D) 0 (E) 54. Multiple Choice On what interval is the function g() e decreasing? (A) (, ] (B) [0, 4] (C) [, 4] (D) (4, ) (E) no interval 55. Multiple Choice Which of the following functions is an antiderivative of? (A) 3 (B) (C) (D) 5 (E) Multiple Choice All of the following functions satisf the conditions of the Mean Value Theorem on the interval [, ] ecept (A) sin (B) sin (C) 5/3 (D) 3/5 (E) Eplorations 57. Analzing Derivative Data Assume that f is continuous on, and differentiable on,. The table gives some values of f (. f ( f ( (a) Estimate where f is increasing, decreasing, and has local etrema. (b) Find a quadratic regression equation for the data in the table and superimpose its graph on a scatter plot of the data. (c) Use the model in part (b) for f and find a formula for f that satisfies f Analzing Motion Data Pria s distance D in meters from a motion detector is given b the data in Table 4.. Table 4. Motion Detector Data t (sec) D (m) t (sec) D (m) (a) Estimate when Pria is moving toward the motion detector; awa from the motion detector. (b) Writing to Learn Give an interpretation of an local etreme values in terms of this problem situation. (c) Find a cubic regression equation D f t for the data in Table 4. and superimpose its graph on a scatter plot of the data. (d) Use the model in (c) for f to find a formula for f. Use this formula to estimate the answers to (a). Etending the Ideas 59. Geometric Mean The geometric mean of two positive numbers a and b is a b. Show that for f on an interval a, b of positive numbers, the value of c in the conclusion of the Mean Value Theorem is c a b. 60. Arithmetic Mean The arithmetic mean of two numbers a and b is a b. Show that for f on an interval a, b, the value of c in the conclusion of the Mean Value Theorem is c a b. 6. Upper Bounds Show that for an numbers a and b, sin b sin a b a. 6. Sign of f Assume that f is differentiable on a b and that f b f a. Show that f is negative at some point between a and b. 63. Monotonic Functions Show that monotonic increasing and decreasing functions are one-to-one

20 Section 4.3 Connecting f and f with the Graph of f What ou ll learn about First Derivative Test for Local Etrema Concavit Points of Inflection Second Derivative Test for Local Etrema Learning about Functions from Derivatives... and wh Differential calculus is a powerful problem-solving tool precisel because of its usefulness for analzing functions. Connecting f and f with the Graph of f First Derivative Test for Local Etrema As we see once again in Figure 4.8, a function f ma have local etrema at some critical points while failing to have local etrema at others. The ke is the sign of f in a critical point s immediate vicinit. As moves from left to right, the values of f increase where f 0 and decrease where f 0. At the points where f has a minimum value, we see that f 0 on the interval immediatel to the left and f 0 on the interval immediatel to the right. (If the point is an endpoint, there is onl the interval on the appropriate side to consider.) This means that the curve is falling (values decreasing) on the left of the minimum value and rising (values increasing) on its right. Similarl, at the points where f has a maimum value, f 0 on the interval immediatel to the left and f 0 on the interval immediatel to the right. This means that the curve is rising (values increasing) on the left of the maimum value and falling (values decreasing) on its right. No etreme f' 0 Local ma f' 0 f' 0 f' 0 f() Absolute ma f' undefined f' 0 f' 0 No etreme f' 0 f' 0 f' 0 Local min f' 0 Local min Absolute min a c c c 3 c 4 c 5 b Figure 4.8 A function s first derivative tells how the graph rises and falls. THEOREM 4 First Derivative Test for Local Etrema The following test applies to a continuous function f. At a critical point c:. If f changes sign from positive to negative at c f 0 for c and f 0 for c, then f has a local maimum value at c. local ma local ma f' 0 f' 0 f' 0 f' 0 c (a) f'(c) 0 c (b) f'(c) undefined continued

21 06 Chapter 4 Applications of Derivatives. If f changes sign from negative to positive at c f 0 for c and f 0 for c, then f has a local minimum value at c. f' 0 local min f' 0 f' 0 local min f' 0 c (a) f'(c) 0 c (b) f'(c) undefined 3. If f does not change sign at c f has the same sign on both sides of c, then f has no local etreme value at c. no etreme f' 0 no etreme f' 0 f' 0 f ' 0 c (a) f'(c) 0 c (b) f'(c) undefined At a left endpoint a: If f 0(f 0) for a, then f has a local maimum (minimum) value at a. local ma f' 0 local min f' 0 a a At a right endpoint b: If f 0(f 0) for b, then f has a local minimum (maimum) value at b. local ma f' 0 local min f' 0 b b Here is how we appl the First Derivative Test to find the local etrema of a function. The critical points of a function f partition the -ais into intervals on which f is either positive or negative. We determine the sign of f in each interval b evaluating f for one value of in the interval. Then we appl Theorem 4 as shown in Eamples and. EXAMPLE Using the First Derivative Test For each of the following functions, use the First Derivative Test to find the local etreme values. Identif an absolute etrema. (a) f () 3 5 (b) g() ( 3)e continued

22 Section 4.3 Connecting f and f with the Graph of f 07 (a) Since f is differentiable for all real numbers, the onl possible critical points are the zeros of f. Solving f () 3 0, we find the zeros to be and. The zeros partition the -ais into three intervals, as shown below: Sign of f + + [ 5, 5] b [ 5, 5] Figure 4.9 The graph of f 3 5. Using the First Derivative Test, we can see from the sign of f on each interval that there is a local maimum at and a local minimum at. The local maimum value is f ( ), and the local minimum value is f(). There are no absolute etrema, as the function has range (, ) (Figure 4.9). (b) Since g is differentiable for all real numbers, the onl possible critical points are the zeros of g. Since g () ( 3) e () e ( 3) e, we find the zeros of g to be and 3. The zeros partition the -ais into three intervals, as shown below: Sign of g [ 5, 5] b [ 8, 5] Figure 4.0 The graph of g 3 e. Using the First Derivative Test, we can see from the sign of f on each interval that there is a local maimum at 3 and a local minimum at. The local maimum value is g( 3) 6e , and the local minimum value is g() e Although this function has the same increasing decreasing increasing pattern as f, its left end behavior is quite different. We see that lim g() 0, so the graph approaches the -ais asmptoticall and is therefore bounded below. This makes g() an absolute minimum. Since lim g(), there is no absolute maimum (Figure 4.0). Now tr Eercise 3. ' decreases CONCAVE DOWN 0 3 CONCAVE UP ' increases Concavit As ou can see in Figure 4., the function 3 rises as increases, but the portions defined on the intervals, 0 and 0, turn in different was. Looking at tangents as we scan from left to right, we see that the slope of the curve decreases on the interval, 0 and then increases on the interval 0,. The curve 3 is concave down on, 0 and concave up on 0,. The curve lies below the tangents where it is concave down, and above the tangents where it is concave up. Figure 4. The graph of 3 is concave down on, 0 and concave up on 0,. DEFINITION Concavit The graph of a differentiable function f () is (a) concave up on an open interval I if is increasing on I. (b) concave down on an open interval I if is decreasing on I. If a function f has a second derivative, then we can conclude that increases if 0 and decreases if 0.

23 08 Chapter 4 Applications of Derivatives CONCAVE UP 4 3 '' > 0 '' > 0 0 CONCAVE UP Figure 4. The graph of is concave up on an interval. (Eample ) 3 sin, sin Concavit Test The graph of a twice-differentiable function f () is (a) concave up on an interval where 0. (b) concave down on an interval where 0. EXAMPLE Determining Concavit Use the Concavit Test to determine the concavit of the given functions on the given intervals: (a) on (3, 0) (b) 3 sin on (0, p) (a) Since is alwas positive, the graph of is concave up on an interval. In particular, it is concave up on (3, 0) (Figure 4.). (b) The graph of 3 sin is concave down on (0, p), where sin is negative. It is concave up on (p,p), where sin is positive (Figure 4.3). Now tr Eercise 7. Points of Inflection The curve 3 sin in Eample changes concavit at the point p, 3. We call p, 3 a point of inflection of the curve. [0, p] b [, 5] Figure 4.3 Using the graph of to determine the concavit of. (Eample ) DEFINITION Point of Inflection A point where the graph of a function has a tangent line and where the concavit changes is a point of inflection. A point on a curve where is positive on one side and negative on the other is a point of inflection. At such a point, is either zero (because derivatives have the intermediate value propert) or undefined. If is a twice differentiable function, 0 at a point of inflection and has a local maimum or minimum. X= Y= [, ] b [, ] Figure 4.4 Graphical confirmation that the graph of e has a point of inflection at / (and hence also at / ). (Eample 3) EXAMPLE 3 Finding Points of Inflection Find all points of inflection of the graph of e. First we find the second derivative, recalling the Chain and Product Rules: e e ( ) e ( ) ( ) e ( ) e (4 ) The factor e is alwas positive, while the factor (4 ) changes sign at / and at /. Since must also change sign at these two numbers, the points of inflection are ( /, / e ) and ( /, / e ). We confirm our solution graphicall b observing the changes of curvature in Figure 4.4. Now tr Eercise 3.

24 Section 4.3 Connecting f and f with the Graph of f 09 Figure 4.6 A possible graph of f. (Eample 4) Figure 4.5 The graph of f, the derivative of f, on the interval [ 4, 4] EXAMPLE 4 Reading the Graph of the Derivative The graph of the derivative of a function f on the interval [ 4, 4] is shown in Figure 4.5. Answer the following questions about f, justifing each answer with information obtained from the graph of f. (a) On what intervals is f increasing? (b) On what intervals is the graph of f concave up? (c) At which -coordinates does f have local etrema? (d) What are the -coordinates of all inflection points of the graph of f? (e) Sketch a possible graph of f on the interval [ 4, 4]. (a) Since f 0 on the intervals [ 4, ) and (, ), the function f must be increasing on the entire interval [ 4, ] with a horizontal tangent at (a shelf point ). (b) The graph of f is concave up on the intervals where f is increasing. We see from the graph that f is increasing on the intervals (, 0) and (3, 4). (c) B the First Derivative Test, there is a local maimum at because the sign of f changes from positive to negative there. Note that there is no etremum at, since f does not change sign. Because the function increases from the left endpoint and decreases to the right endpoint, there are local minima at the endpoints 4 and 4. (d) The inflection points of the graph of f have the same -coordinates as the turning points of the graph of f, namel, 0, and 3. (e) A possible graph satisfing all the conditions is shown in Figure 4.6. Now tr Eercise 3. [ 4.7, 4.7] b [ 3., 3.] Figure 4.7 The function f () 4 does not have a point of inflection at the origin, even though f (0) 0. Caution: It is tempting to oversimplif a point of inflection as a point where the second derivative is zero, but that can be wrong for two reasons:. The second derivative can be zero at a noninflection point. For eample, consider the function f () 4 (Figure 4.7). Since f (), we have f (0) 0; however, (0, 0) is not an inflection point. Note that f does not change sign at 0.. The second derivative need not be zero at an inflection point. For eample, consider the function f () 3 (Figure 4.8). The concavit changes at 0, but there is a vertical tangent line, so both f (0) and f (0) fail to eist. Therefore, the onl safe wa to test algebraicall for a point of inflection is to confirm a sign change of the second derivative. This could occur at a point where the second derivative is zero, but it also could occur at a point where the second derivative fails to eist. To stud the motion of a bod moving along a line, we often graph the bod s position as a function of time. One reason for doing so is to reveal where the bod s acceleration, given b the second derivative, changes sign. On the graph, these are the points of inflection. [ 4.7, 4.7] b [ 3., 3.] Figure 4.8 The function f () 3 has a point of inflection at the origin, even though f (0) 0. EXAMPLE 5 Studing Motion along a Line A particle is moving along the -ais with position function t t 3 4t t 5, t 0. Find the velocit and acceleration, and describe the motion of the particle. continued

25 0 Chapter 4 Applications of Derivatives Solve Analticall The velocit is and the acceleration is v t t 6t 8t t 3t, [0, 6] b [ 30, 30] (a) a t v t t t 8 4 3t 7. When the function t is increasing, the particle is moving to the right on the -ais; when t is decreasing, the particle is moving to the left. Figure 4.9 shows the graphs of the position, velocit, and acceleration of the particle. Notice that the first derivative (v ) is zero when t and t /3. These zeros partition the t-ais into three intervals, as shown in the sign graph of v below: Sign of v = ' [0, 6] b [ 30, 30] (b) Behavior of increasing decreasing increasing Particle motion right left right The particle is moving to the right in the time intervals [0, ) and (/3, ) and moving to the left in (, /3). The acceleration a(t) t 8 has a single zero at t 7/3. The sign graph of the acceleration is shown below: Sign of a = " + [0, 6] b [ 30, 30] (c) Figure 4.9 The graph of (a) t t 3 4t t 5, t 0, (b) t 6t 8t, and (c) t t 8. (Eample 5) Graph of Particle motion 0 concave down decelerating 7 3 concave up accelerating The accelerating force is directed toward the left during the time interval [0, 7/3], is momentaril zero at t 7/3, and is directed toward the right thereafter. Now tr Eercise 5. 5 Point of inflection Figure 4.30 A logistic curve. a ce b The growth of an individual compan, of a population, in sales of a new product, or of salaries often follows a logistic or life ccle curve like the one shown in Figure For eample, sales of a new product will generall grow slowl at first, then eperience a period of rapid growth. Eventuall, sales growth slows down again. The function f in Figure 4.30 is increasing. Its rate of increase, f, is at first increasing f 0 up to the point of inflection, and then its rate of increase, f, is decreasing f 0. This is, in a sense, the opposite of what happens in Figure 4.. Some graphers have the logistic curve as a built-in regression model. We use this feature in Eample 6.

26 Section 4.3 Connecting f and f with the Graph of f Table 4. Population of Alaska Years since 900 Population 0 55, , , , , , , , ,93 Source: Bureau of the Census, U.S. Chamber of Commerce. [, 08] b [0, ] (a) Zero X= Y=0 [, 08] b [ 50, 50] (b) Figure 4.3 (a) The logistic regression curve e superimposed on the population data from Table 4., and (b) the graph of showing a zero at about 83. EXAMPLE 6 Population Growth in Alaska Table 4. shows the population of Alaska in each 0-ear census between 90 and 000. (a) Find the logistic regression for the data. (b) Use the regression equation to predict the Alaskan population in the 00 census. (c) Find the inflection point of the regression equation. What significance does the inflection point have in terms of population growth in Alaska? (d) What does the regression equation indicate about the population of Alaska in the long run? (a) Using ears since 900 as the independent variable and population as the dependent variable, the logistic regression equation is approimatel e Its graph is superimposed on a scatter plot of the data in Figure 4.3(a). Store the regression equation as Y in our calculator. (b) The calculator reports Y(0) to be approimatel 78,53. (Given the uncertaint of this kind of etrapolation, it is probabl more reasonable to sa approimatel 78,00. ) (c) The inflection point will occur where changes sign. Finding algebraicall would be tedious, but we can graph the numerical derivative of the numerical derivative and find the zero graphicall. Figure 4.3(b) shows the graph of, which is nderiv(nderiv(y,x,x),x,x) in calculator snta. The zero is approimatel 83, so the inflection point occurred in 983, when the population was about 450,570 and growing the fastest. (d) Notice that lim , so the regression equation 7.57e indicates that the population of Alaska will stabilize at about 895,600 in the long run. Do not put too much faith in this number, however, as human population is dependent on too man variables that can, and will, change over time. Now tr Eercise 3. Second Derivative Test for Local Etrema Instead of looking for sign changes in at critical points, we can sometimes use the following test to determine the presence of local etrema. THEOREM 5 Second Derivative Test for Local Etrema. If f c 0 and f c 0, then f has a local maimum at c.. If f c 0 and f c 0, then f has a local minimum at c. This test requires us to know f onl at c itself and not in an interval about c. This makes the test eas to appl. That s the good news. The bad news is that the test fails if f c 0 or if f c fails to eist. When this happens, go back to the First Derivative Test for local etreme values. In Eample 7, we appl the Second Derivative Test to the function in Eample.

27 Chapter 4 Applications of Derivatives EXAMPLE 7 Using the Second Derivative Test Find the local etreme values of f 3 5. We have f 3 3( 4) f 6. Testing the critical points (there are no endpoints), we find f 0 f has a local maimum at and f 0 f has a local minimum at. Now tr Eercise 35. EXAMPLE 8 Using f and f to Graph f Let f 4 3. (a) Identif where the etrema of f occur. (b) Find the intervals on which f is increasing and the intervals on which f is decreasing. (c) Find where the graph of f is concave up and where it is concave down. (d) Sketch a possible graph for f. f is continuous since f eists. The domain of f is,, so the domain of f is also,. Thus, the critical points of f occur onl at the zeros of f. Since f , the first derivative is zero at 0 and 3. Intervals Sign of f Behavior of f decreasing decreasing increasing (a) Using the First Derivative Test and the table above we see that there is no etremum at 0 and a local minimum at 3. (b) Using the table above we see that f is decreasing in, 0 and 0, 3, and increasing in 3,. Note The Second Derivative Test does not appl at 0 because f 0 0. We need the First Derivative Test to see that there is no local etremum at 0. (c) f 4 is zero at 0 and. Intervals 0 0 Sign of f Behavior of f concave up concave down concave up We see that f is concave up on the intervals, 0 and,, and concave down on 0,. continued

28 Section 4.3 Connecting f and f with the Graph of f 3 (d) Summarizing the information in the two tables above we obtain decreasing decreasing decreasing increasing concave up concave down concave up concave up Figure 4.3 The graph for f has no etremum but has points of inflection where 0 and, and a local minimum where 3. (Eample 8) Figure 4.3 shows one possibilit for the graph of f. Now tr Eercise 39. EXPLORATION Let f 4 3. Finding f from f. Find three different functions with derivative equal to f. How are the graphs of the three functions related?. Compare their behavior with the behavior found in Eample 8. Learning about Functions from Derivatives We have seen in Eample 8 and Eploration that we are able to recover almost everthing we need to know about a differentiable function f b eamining. We can find where the graph rises and falls and where an local etrema are assumed. We can differentiate to learn how the graph bends as it passes over the intervals of rise and fall. We can determine the shape of the function s graph. The onl information we cannot get from the derivative is how to place the graph in the -plane. As we discovered in Section 4., the onl additional information we need to position the graph is the value of f at one point. f() f() f() Differentiable smooth, connected; graph ma rise and fall ' 0 graph rises from left to right; ma be wav ' 0 graph falls from left to right; ma be wav or or '' 0 concave up throughout; no waves; graph ma rise or fall '' 0 concave down throughout; no waves; graph ma rise or fall '' changes sign Inflection point or ' changes sign graph has local maimum or minimum ' 0 and '' 0 at a point; graph has local maimum ' 0 and '' 0 at a point; graph has local minimum

29 4 Chapter 4 Applications of Derivatives Remember also that a function can be continuous and still have points of nondifferentiabilit (cusps, corners, and points with vertical tangent lines). Thus, a noncontinuous graph of f could lead to a continuous graph of f, as Eample 9 shows. EXAMPLE 9 Analzing a Discontinuous Derivative A function f is continuous on the interval [ 4, 4]. The discontinuous function f, with domain [ 4, 0) (0, ) (, 4], is shown in the graph to the right (Figure 4.33). (a) Find the -coordinates of all local etrema and points of inflection of f. (b) Sketch a possible graph of f. Figure 4.33 The graph of f, a discontinuous derivative (a) For etrema, we look for places where f changes sign. There are local maima at 3, 0, and (where f goes from positive to negative) and local minima at and (where f goes from negative to positive). There are also local minima at the two endpoints 4 and 4, because f starts positive at the left endpoint and ends negative at the right endpoint. For points of inflection, we look for places where f changes sign, that is, where the graph of f changes direction. This occurs onl at. (b) A possible graph of f is shown in Figure The derivative information determines the shape of the three components, and the continuit condition determines that the three components must be linked together. Now tr Eercises 49 and 53. Figure 4.34 A possible graph of f. (Eample 9) EXPLORATION Finding f from f and f A function f is continuous on its domain, 4, f 5, f 4, and f and f have the following properties f does not eist 0 f does not eist 0. Find where all absolute etrema of f occur.. Find where the points of inflection of f occur. 3. Sketch a possible graph of f. Quick Review 4.3 (For help, go to Sections.3,., 3.3, and 3.9.) In Eercises and, factor the epression and use sign charts to solve the inequalit In Eercises 3 6, find the domains of f and f. 3. f e 4. f f 6. f 5 In Eercises 7 0, find the horizontal asmptotes of the function s graph e 8. e e 0.5 5e 0.

30 Section 4.3 Connecting f and f with the Graph of f 5 Section 4.3 Eercises In Eercises 6, use the First Derivative Test to determine the local etreme values of the function, and identif an absolute etrema. Support our answers graphicall e In Eercises 3 and 4, use the graph of the function f to estimate the intervals on which the function f is (a) increasing or (b) decreasing. Also, (c) estimate the -coordinates of all local etreme values. 3. f '() , 0 6 {, 0 0 In Eercises 7, use the Concavit Test to determine the intervals on which the graph of the function is (a) concave up and (b) concave down = f '(),. {,. e, 0 p 0 In Eercises 3 0, find all points of inflection of the function. 3. e tan In Eercises and, use the graph of the function f to estimate where (a) f and (b) f are 0, positive, and negative.. = f() In Eercises 5 8, a particle is moving along the -ais with position function (t). Find the (a) velocit and (b) acceleration, and (c) describe the motion of the particle for t t t 4t 3 6. t 6 t t 7. t t 3 3t 3 8. t 3t t 3 In Eercises 9 and 30, the graph of the position function s t of a particle moving along a line is given. At approimatel what times is the particle s (a) velocit equal to zero? (b) acceleration equal to zero? 9. 0 Distance from origin = s(t) 0 5 Time (sec) 0 5 t. = f() Distance from origin 0 5 s(t) Time (sec) 0 5 t

31 6 Chapter 4 Applications of Derivatives 3. Table 4.3 shows the population of Pennslvania in each 0-ear census between 830 and 950. Table 4.3 Years since 80 Population of Pennslvania (a) Find the logistic regression for the data. Population in thousands ,498 Source: Bureau of the Census, U.S. Chamber of Commerce. (b) Graph the data in a scatter plot and superimpose the regression curve. (c) Use the regression equation to predict the Pennslvania population in the 000 census. (d) In what ear was the Pennslvania population growing the fastest? What significant behavior does the graph of the regression equation ehibit at that point? (e) What does the regression equation indicate about the population of Pennslvania in the long run? 3. In 977, there were,68,450 basic cable television subscribers in the U.S. Table 4.4 shows the cumulative number of subscribers added to that baseline number from 978 to 985. (c) In what ear between 977 and 985 were basic cable TV subscriptions growing the fastest? What significant behavior does the graph of the regression equation ehibit at that point? (d) What does the regression equation indicate about the number of basic cable television subscribers in the long run? (Be sure to add the baseline 977 number.) (e) Writing to Learn In fact, the long-run number of basic cable subscribers predicted b the regression equation falls short of the actual 00 number b more than 3 million. What circumstances changed to render the earlier model so ineffective? In Eercises 33 38, use the Second Derivative Test to find the local etrema for the function e 38. e In Eercises 39 and 40, use the derivative of the function f to find the points at which f has a (a) local maimum, (b) local minimum, or (c) point of inflection Eercises 4 and 4 show the graphs of the first and second derivatives of a function f. Cop the figure and add a sketch of a possible graph of f that passes through the point P. 4. Table 4.4 Growth of Cable Television Added Subscribers Years since 977 since 977,39,90,84, ,67,490 4,9,00 5 7,340,570 6,3, ,90, ,87,50 Source: Nielsen Media Research, as reported in The World Almanac and Book of Facts f'() f''() P P f'() (a) Find the logistic regression for the data. (b) Graph the data in a scatter plot and superimpose the regression curve. Does it fit the data well? O f''()

32 Section 4.3 Connecting f and f with the Graph of f Writing to Learn If f is a differentiable function and f c 0 at an interior point c of f s domain, must f have a local maimum or minimum at c? Eplain. 44. Writing to Learn If f is a twice-differentiable function and f c 0 at an interior point c of f s domain, must f have an inflection point at c? Eplain. 45. Connecting f and f Sketch a smooth curve f through the origin with the properties that f 0 for 0 and f 0 for Connecting f and f Sketch a smooth curve f through the origin with the properties that f 0 for 0 and f 0 for Connecting f, f, and f Sketch a continuous curve f with the following properties. Label coordinates where possible. f 8 f 0 for f 0 4 f 0 for f 0 f 0 for 0 f f 0 f 0 for Using Behavior to Sketch Sketch a continuous curve f with the following properties. Label coordinates where possible. Curve falling, concave up horizontal tangent 4 rising, concave up 4 4 inflection point 4 6 rising, concave down 6 7 horizontal tangent 6 falling, concave down In Eercises 49 and 50, use the graph of f to estimate the intervals on which the function f is (a) increasing or (b) decreasing. Also, (c) estimate the -coordinates of all local etreme values. (Assume that the function f is continuous, even at the points where f is undefined.) 49. The domain of f is 0, 4 4, The domain of f is 0,,, 3. = f'() 3 Group Activit In Eercises 5 and 5, do the following. (a) Find the absolute etrema of f and where the occur. (b) Find an points of inflection. (c) Sketch a possible graph of f. 5. f is continuous on 0, 3 and satisfies the following. 0 3 f 0 0 f 3 0 does not eist 3 f 0 does not eist f f f 5. f is an even function, continuous on 3, 3, and satisfies the following. 0 f 0 f does not eist 0 does not eist f does not eist 0 does not eist 0 3 f f f (d) What can ou conclude about f 3 and f 3? = f'() Group Activit In Eercises 53 and 54, sketch a possible graph of a continuous function f that has the given properties. 53. Domain 0, 6, graph of f given in Eercise 49, and f Domain 0, 3, graph of f given in Eercise 50, and f 0 3.

33 8 Chapter 4 Applications of Derivatives Standardized Test Questions You should solve the following problems without using a graphing calculator. 55. True or False If f c 0, then (c, f (c)) is a point of inflection. Justif our answer. 56. True or False If f c 0 and f c 0, then f (c) is a local maimum. Justif our answer. 57. Multiple Choice If a 0, the graph of a is concave up on (A), a (B), a (C) a, (D) a, (E), 58. Multiple Choice If f 0 f 0 f 0 0, which of the following must be true? (A) There is a local maimum of f at the origin. (B) There is a local minimum of f at the origin. (C) There is no local etremum of f at the origin. (D) There is a point of inflection of the graph of f at the origin. (E) There is a horizontal tangent to the graph of f at the origin. 59. Multiple Choice The -coordinates of the points of inflection of the graph of are (A) 0 onl (B) onl (C) 3 onl (D) 0 and 3 (E) 0 and 60. Multiple Choice Which of the following conditions would enable ou to conclude that the graph of f has a point of inflection at c? (A) There is a local maimum of f at c. (B) f c 0. (C) f c does not eist. (D) The sign of f changes at c. (E) f is a cubic polnomial and c 0. Eploration 6. Graphs of Cubics There is almost no leewa in the locations of the inflection point and the etrema of f a 3 b c d, a 0, because the one inflection point occurs at b 3a and the etrema, if an, must be located smmetricall about this value of. Check this out b eamining (a) the cubic in Eercise 7 and (b) the cubic in Eercise. Then (c) prove the general case. Etending the Ideas In Eercises 6 and 63, feel free to use a CAS (computer algebra sstem), if ou have one, to solve the problem. 6. Logistic Functions Let f c ae b with a 0, abc 0. (a) Show that f is increasing on the interval, if abc 0, and decreasing if abc 0. (b) Show that the point of inflection of f occurs at ln a b. 63. Quartic Polnomial Functions Let f a 4 b 3 c d e with a 0. (a) Show that the graph of f has 0 or points of inflection. (b) Write a condition that must be satisfied b the coefficients if the graph of f has 0 or points of inflection. Quick Quiz for AP* Preparation: Sections You should solve these problems without using a graphing calculator.. Multiple Choice How man critical points does the function f () ( ) 5 ( 3) 4 have? (A) One (B) Two (C) Three (D) Five (E) Nine. Multiple Choice For what value of does the function f () ( ) ( 3) have a relative maimum? (A) 3 (B) 7 3 (C) 5 (D) 7 3 (E) 5 3. Multiple Choice If g is a differentiable function such that g() 0 for all real numbers, and if f () ( 9)g(), which of the following is true? (A) f has a relative maimum at 3 and a relative minimum at 3. (B) f has a relative minimum at 3 and a relative maimum at 3. (C) f has relative minima at 3 and at 3. (D) f has relative maima at 3 and at 3. (E) It cannot be determined if f has an relative etrema. 4. Free Response Let f be the function given b f 3 ln ( ) with domain [, 4]. (a) Find the coordinate of each relative maimum point and each relative minimum point of f. Justif our answer. (b) Find the -coordinate of each point of inflection of the graph of f. (c) Find the absolute maimum value of f.

34 Section 4.4 Modeling and Optimization What ou ll learn about Eamples from Mathematics Eamples from Business and Industr Eamples from Economics Modeling Discrete Phenomena with Differentiable Functions... and wh Historicall, optimization problems were among the earliest applications of what we now call differential calculus. Modeling and Optimization Eamples from Mathematics While toda s graphing technolog makes it eas to find etrema without calculus, the algebraic methods of differentiation were understandabl more practical, and certainl more accurate, when graphs had to be rendered b hand. Indeed, one of the oldest applications of what we now call differential calculus (pre-dating Newton and Leibniz) was to find maimum and minimum values of functions b finding where horizontal tangent lines might occur. We will use both algebraic and graphical methods in this section to solve ma-min problems in a variet of contets, but the emphasis will be on the modeling process that both methods have in common. Here is a strateg ou can use: Strateg for Solving Ma-Min Problems. Understand the Problem Read the problem carefull. Identif the information ou need to solve the problem.. Develop a Mathematical Model of the Problem Draw pictures and label the parts that are important to the problem. Introduce a variable to represent the quantit to be maimized or minimized. Using that variable, write a function whose etreme value gives the information sought. 3. Graph the Function Find the domain of the function. Determine what values of the variable make sense in the problem. 4. Identif the Critical Points and Endpoints Find where the derivative is zero or fails to eist. 5. Solve the Mathematical Model If unsure of the result, support or confirm our solution with another method. 6. Interpret the Solution Translate our mathematical result into the problem setting and decide whether the result makes sense. [ 5, 5] b [ 00, 50] Figure 4.35 The graph of f 0 with domain, has an absolute maimum of 00 at 0. (Eample ) EXAMPLE Using the Strateg Find two numbers whose sum is 0 and whose product is as large as possible. Model If one number is, the other is 0, and their product is f 0. Solve Graphicall We can see from the graph of f in Figure 4.35 that there is a maimum. From what we know about parabolas, the maimum occurs at 0. P Q Interpret The two numbers we seek are 0 and 0 0. Now tr Eercise. Sometimes we find it helpful to use both analtic and graphical methods together, as in Eample. [0, π] b [ 0.5,.5] Figure 4.36 A rectangle inscribed under one arch of sin. (Eample ) EXAMPLE Inscribing Rectangles A rectangle is to be inscribed under one arch of the sine curve (Figure 4.36). What is the largest area the rectangle can have, and what dimensions give that area? continued

35 0 Chapter 4 Applications of Derivatives Model Let, sin be the coordinates of point P in Figure From what we know about the sine function the -coordinate of point Q is p. Thus, Maimum X = Y =.97 [0, p/] b [, ] (a) Zero X = Y = 0 [0, p/] b [ 4, 4] (b) Figure 4.37 The graph of (a) A p sin and (b) A in the interval 0 p. (Eample ) and The area of the rectangle is p length of rectangle sin height of rectangle. A p sin. Solve Analticall and Graphicall We can assume that 0 p. Notice that A 0 at the endpoints 0 and p. Since A is differentiable, the onl critical points occur at the zeros of the first derivative, A sin p cos. It is not possible to solve the equation A 0 using algebraic methods. We can use the graph of A (Figure 4.37a) to find the maimum value and where it occurs. Or, we can use the graph of A (Figure 4.37b) to find where the derivative is zero, and then evaluate A at this value of to find the maimum value. The two -values appear to be the same, as the should. Interpret The rectangle has a maimum area of about. square units when 0.7. At this point, the rectangle is p.7 units long b sin 0.65 unit high. Now tr Eercise 5. EXPLORATION Constructing Cones A cone of height h and radius r is constructed from a flat, circular disk of radius 4 in. b removing a sector AOC of arc length in. and then connecting the edges OA and OC. What arc length will produce the cone of maimum volume, and what is that volume? A O 4" 4" C O A h 4". Show that r 8p, h 6 r p, and V p 3 ( 8p p C ) 6 ( 8p p ). r NOT TO SCALE. Show that the natural domain of V is 0 6p. Graph V over this domain. 3. Eplain wh the restriction 0 8p makes sense in the problem situation. Graph V over this domain. 4. Use graphical methods to find where the cone has its maimum volume, and what that volume is. 5. Confirm our findings in part 4 analticall. [Hint: Use V 3 pr h, h r 6, and the Chain Rule.]

36 Section 4.4 Modeling and Optimization Eamples from Business and Industr 0" To optimize something means to maimize or minimize some aspect of it. What is the size of the most profitable production run? What is the least epensive shape for an oil can? What is the stiffest rectangular beam we can cut from a -inch log? We usuall answer such questions b finding the greatest or smallest value of some function that we have used to model the situation. 5" (a) EXAMPLE 3 Fabricating a Bo An open-top bo is to be made b cutting congruent squares of side length from the corners of a 0- b 5-inch sheet of tin and bending up the sides (Figure 4.38). How large should the squares be to make the bo hold as much as possible? What is the resulting maimum volume? 5 Model The height of the bo is, and the other two dimensions are 0 and 5. Thus, the volume of the bo is 0 (b) Figure 4.38 An open bo made b cutting the corners from a piece of tin. (Eample 3) (0 )(5 ) V 0 5. Solve Graphicall Because cannot eceed 0, we have 0 0. Figure 4.39 suggests that the maimum value of V is about and occurs at Confirm Analticall Epanding, we obtain V The first derivative of V is V The two solutions of the quadratic equation V 0 are c and Maimum X = Y = [0, 0] b [ 300, 000] Figure 4.39 We chose the 300 in so that the coordinates of the local maimum at the bottom of the screen would not interfere with the graph. (Eample 3) c 4.3. Onl c is in the domain 0, 0 of V. The values of V at this one critical point and the two endpoints are Critical point value: V c Endpoint values: V 0 0, V 0 0. Interpret Cutout squares that are about 3.68 in. on a side give the maimum volume, about in 3. Now tr Eercise 7. EXAMPLE 4 Designing a Can You have been asked to design a one-liter oil can shaped like a right circular clinder (see Figure 4.40 on the net page). What dimensions will use the least material? continued

37 Chapter 4 Applications of Derivatives r Figure 4.40 This one-liter can uses the least material when h r. (Eample 4) h Volume of can: If r and h are measured in centimeters, then the volume of the can in cubic centimeters is Surface area of can: pr h 000. liter 000 cm 3 A pr prh circular clinder ends wall How can we interpret the phrase least material? One possibilit is to ignore the thickness of the material and the waste in manufacturing. Then we ask for dimensions r and h that make the total surface area as small as possible while satisfing the constraint pr h 000. (Eercise 7 describes one wa to take waste into account.) Model To epress the surface area as a function of one variable, we solve for one of the variables in pr h 000 and substitute that epression into the surface area formula. Solving for h is easier, h 000. pr Thus, A pr prh pr pr( 000 pr ) pr r Solve Analticall Our goal is to find a value of r 0 that minimizes the value of A. Figure 4.4 suggests that such a value eists. Notice from the graph that for small r (a tall thin container, like a piece of pipe), the term 000 r dominates and A is large. For large r (a short wide container, like a pizza pan), the term pr dominates and A again is large. Since A is differentiable on r 0, an interval with no endpoints, it can have a minimum value onl where its first derivative is zero. [0, 5] b [0, 000] Figure 4.4 The graph of A pr 000 r, r 0. (Eample 4) d A 4pr 0 00 dr r 0 4pr 0 00 r Set da dr 0. 4pr Multipl b r. r p Solve for r. Something happens at r p, but what? If the domain of A were a closed interval, we could find out b evaluating A at this critical point and the endpoints and comparing the results. But the domain is an open interval, so we must learn what is happening at r p b referring to the shape of A s graph. The second derivative d A dr 4p r 3 is positive throughout the domain of A. The graph is therefore concave up and the value of A at r p an absolute minimum. continued

38 Section 4.4 Modeling and Optimization 3 The corresponding value of h (after a little algebra) is h 000 pr p 0 r. Interpret The one-liter can that uses the least material has height equal to the diameter, with r 5.4 cm and h 0.84 cm. Now tr Eercise. Marginal Analsis Because differentiable functions are locall linear, we can use the marginals to approimate the etra revenue, cost, or profit resulting from selling or producing one more item. Using these approimations is referred to as marginal analsis. Eamples from Economics Here we want to point out two more places where calculus makes a contribution to economic theor. The first has to do with maimizing profit. The second has to do with minimizing average cost. Suppose that r the revenue from selling items, c the cost of producing the items, p r c the profit from selling items. The marginal revenue, marginal cost, and marginal profit at this production level ( items) are dr dc marginal revenue, marginal cost, d p marginal profit. d d d The first observation is about the relationship of p to these derivatives. THEOREM 6 Maimum Profit Maimum profit (if an) occurs at a production level at which marginal revenue equals marginal cost. Proof We assume that r and c are differentiable for all 0, so if p r c has a maimum value, it occurs at a production level at which p 0. Since p r c, p 0 implies that r c 0 or r c. Figure 4.4 gives more information about this situation. Cost c() Figure 4.4 The graph of a tpical cost function starts concave down and later turns concave up. It crosses the revenue curve at the breakeven point B. To the left of B, the compan operates at a loss. To the right, the compan operates at a profit, the maimum profit occurring where r c. Farther to the right, cost eceeds revenue (perhaps because of a combination of market saturation and rising labor and material costs) and production levels become unprofitable again. Dollars 0 Revenue r() Break-even Maimum profit, c'() r'() B Maimum loss (minimum profit), c'() r'() Items produced

39 4 Chapter 4 Applications of Derivatives What guidance do we get from this observation? We know that a production level at which p 0 need not be a level of maimum profit. It might be a level of minimum profit, for eample. But if we are making financial projections for our compan, we should look for production levels at which marginal cost seems to equal marginal revenue. If there is a most profitable production level, it will be one of these. EXAMPLE 5 Maimizing Profit Suppose that r 9 and c 3 6 5, where represents thousands of units. Is there a production level that maimizes profit? If so, what is it? Notice that r 9 and c The two solutions of the quadratic equation are Set c r and c() r() 9 Maimum for profit The possible production levels for maimum profit are thousand units or 3.44 thousand units. The graphs in Figure 4.43 show that maimum profit occurs at about 3.44 and maimum loss occurs at about Another wa to look for optimal production levels is to look for levels that minimize the average cost of the units produced. Theorem 7 helps us find them. Now tr Eercise 3. Local maimum for loss 0 NOT TO SCALE Figure 4.43 The cost and revenue curves for Eample 5. THEOREM 7 Minimizing Average Cost The production level (if an) at which average cost is smallest is a level at which the average cost equals the marginal cost. Proof We assume that c is differentiable. c cost of producing items, 0 c average cost of producing items If the average cost can be minimized, it will be a production level at which d d ( c ) 0 c c 0 Quotient Rule c c 0 Multipl b. c c. marginal average cost cost

40 Section 4.4 Modeling and Optimization 5 Again we have to be careful about what Theorem 7 does and does not sa. It does not sa that there is a production level of minimum average cost it sas where to look to see if there is one. Look for production levels at which average cost and marginal cost are equal. Then check to see if an of them gives a minimum average cost. EXAMPLE 6 Minimizing Average Cost Suppose c 3 6 5, where represents thousands of units. Is there a production level that minimizes average cost? If so, what is it? We look for levels at which average cost equals marginal cost. Marginal cost: c 3 5 Average cost: c or 3 Marginal cost Average cost Since 0, the onl production level that might minimize average cost is 3 thousand units. We use the second derivative test. d d ( c ) 6 d d ( c 0 The second derivative is positive for all 0, so 3 gives an absolute minimum. Now tr Eercise 5. ) Modeling Discrete Phenomena with Differentiable Functions In case ou are wondering how we can use differentiable functions c and r to describe the cost and revenue that comes from producing a number of items that can onl be an integer, here is the rationale. When is large, we can reasonabl fit the cost and revenue data with smooth curves c and r that are defined not onl at integer values of but at the values in between just as we do when we use regression equations. Once we have these differentiable functions, which are supposed to behave like the real cost and revenue when is an integer, we can appl calculus to draw conclusions about their values. We then translate these mathematical conclusions into inferences about the real world that we hope will have predictive value. When the do, as is the case with the economic theor here, we sa that the functions give a good model of realit. What do we do when our calculus tells us that the best production level is a value of that isn t an integer, as it did in Eample 5? We use the nearest convenient integer. For 3.44 thousand units in Eample 5, we might use 344, or perhaps 340 or 340 if we ship in boes of 0.

41 6 Chapter 4 Applications of Derivatives Quick Review 4.4 (For help, go to Sections.6, 4., and Appendi A..). Use the first derivative test to identif the local etrema of Use the second derivative test to identif the local etrema of Find the volume of a cone with radius 5 cm and height 8 cm. 4. Find the dimensions of a right circular clinder with volume 000 cm 3 and surface area 600 cm. In Eercises 5 8, rewrite the epression as a trigonometric function of the angle. 5. sin 6. cos 7. sin p 8. cos p In Eercises 9 and 0, use substitution to find the eact solutions of the sstem of equations { { { Section 4.4 Eercises In Eercises 0, solve the problem analticall. Support our answer graphicall.. Finding Numbers The sum of two nonnegative numbers is 0. Find the numbers if (a) the sum of their squares is as large as possible; as small as possible. (b) one number plus the square root of the other is as large as possible; as small as possible.. Maimizing Area What is the largest possible area for a right triangle whose hpotenuse is 5 cm long, and what are its dimensions? 3. Maimizing Perimeter What is the smallest perimeter possible for a rectangle whose area is 6 in, and what are its dimensions? 4. Finding Area Show that among all rectangles with an 8-m perimeter, the one with largest area is a square. 5. Inscribing Rectangles The figure shows a rectangle inscribed in an isosceles right triangle whose hpotenuse is units long. B P(,?) A 0 (a) Epress the -coordinate of P in terms of. [Hint: Write an equation for the line AB.] (b) Epress the area of the rectangle in terms of. (c) What is the largest area the rectangle can have, and what are its dimensions? 6. Largest Rectangle A rectangle has its base on the -ais and its upper two vertices on the parabola. What is the largest area the rectangle can have, and what are its dimensions? 7. Optimal Dimensions You are planning to make an open rectangular bo from an 8- b 5-in. piece of cardboard b cutting congruent squares from the corners and folding up the sides. What are the dimensions of the bo of largest volume ou can make this wa, and what is its volume? 8. Closing Off the First Quadrant You are planning to close off a corner of the first quadrant with a line segment 0 units long running from a, 0 to 0, b. Show that the area of the triangle enclosed b the segment is largest when a b. 9. The Best Fencing Plan A rectangular plot of farmland will be bounded on one side b a river and on the other three sides b a single-strand electric fence. With 800 m of wire at our disposal, what is the largest area ou can enclose, and what are its dimensions? 0. The Shortest Fence A 6-m rectangular pea patch is to be enclosed b a fence and divided into two equal parts b another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?

42 Section 4.4 Modeling and Optimization 7. Designing a Tank Your iron works has contracted to design and build a 500-ft 3, square-based, open-top, rectangular steel holding tank for a paper compan. The tank is to be made b welding thin stainless steel plates together along their edges. As the production engineer, our job is to find dimensions for the base and height that will make the tank weigh as little as possible. (a) What dimensions do ou tell the shop to use? (b) Writing to Learn Briefl describe how ou took weight into account.. Catching Rainwater A 5-ft 3 open-top rectangular tank with a square base ft on a side and ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not onl the material from which the tank is made but also an ecavation charge proportional to the product. (a) If the total cost is c 5 4 0, what values of and will minimize it? (b) Writing to Learn Give a possible scenario for the cost function in (a). 3. Designing a Poster You are designing a rectangular poster to contain 50 in of printing with a 4-in. margin at the top and bottom and a -in. margin at each side. What overall dimensions will minimize the amount of paper used? 8. Designing a Bo with Lid A piece of cardboard measures 0- b 5-in. Two equal squares are removed from the corners of a 0-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular bo with lid. NOT TO SCALE 0" Base 5" Lid (a) Write a formula V for the volume of the bo. (b) Find the domain of V for the problem situation and graph V over this domain. (c) Use a graphical method to find the maimum volume and the value of that gives it. (d) Confirm our result in part (c) analticall. 9. Designing a Suitcase A 4- b 36-in. sheet of cardboard is folded in half to form a 4- b 8-in. rectangle as shown in the figure. Then four congruent squares of side length are cut from the corners of the folded rectangle. The sheet is unfolded, and the si tabs are folded up to form a bo with sides and a lid. 4. Vertical Motion The height of an object moving verticall is given b s 6t 96t, 4" 4" with s in ft and t in sec. Find (a) the object s velocit when t 0, (b) its maimum height and when it occurs, and (c) its velocit when s 0. 36" The sheet is then unfolded. 8" 5. Finding an Angle Two sides of a triangle have lengths a and b, and the angle between them is u. What value of u will maimize the triangle s area? [Hint: A ab sin u.] 6. Designing a Can What are the dimensions of the lightest open-top right circular clindrical can that will hold a volume of 000 cm 3? Compare the result here with the result in Eample 4. 4" Base 7. Designing a Can You are designing a 000-cm 3 right circular clindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure r units on a side. The total amount of aluminum used up b the can will therefore be A 8r prh rather than the A pr prh in Eample 4. In Eample 4 the ratio of h to r for the most economical can was to. What is the ratio now? 36" (a) Write a formula V for the volume of the bo. (b) Find the domain of V for the problem situation and graph V over this domain. (c) Use a graphical method to find the maimum volume and the value of that gives it. (d) Confirm our result in part (c) analticall. (e) Find a value of that ields a volume of 0 in 3. (f) Writing to Learn Write a paragraph describing the issues that arise in part (b).

43 8 Chapter 4 Applications of Derivatives 0. Quickest Route Jane is mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can row mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?. Inscribing Rectangles A rectangle is to be inscribed under the arch of the curve 4 cos 0.5 from p to p. What are the dimensions of the rectangle with largest area, and what is the largest area?. Maimizing Volume Find the dimensions of a right circular clinder of maimum volume that can be inscribed in a sphere of radius 0 cm. What is the maimum volume? 3. Maimizing Profit Suppose r() 8 represents revenue and c() represents cost, with measured in thousands of units. Is there a production level that maimizes profit? If so, what is it? 4. Maimizing Profit Suppose r() /( ) represents revenue and c() ( ) 3 /3 /3 represents cost, with measured in thousands of units. Is there a production level that maimizes profit? If so, what is it? 5. Minimizing Average Cost Suppose c() , where is measured in thousands of units. Is there a production level that minimizes average cost? If so, what is it? 6. Minimizing Average Cost Suppose c() e, where is measured in thousands of units. Is there a production level that minimizes average cost? If so, what is it? 7. Tour Service You operate a tour service that offers the following rates: $00 per person if 50 people (the minimum number to book the tour) go on the tour. For each additional person, up to a maimum of 80 people total, the rate per person is reduced b $. It costs $6000 (a fied cost) plus $3 per person to conduct the tour. How man people does it take to maimize our profit? 8. Group Activit The figure shows the graph of f e, 0. (c) Draw a scatter plot of the data a, A. (d) Find the quadratic, cubic, and quartic regression equations for the data in part (b), and superimpose their graphs on a scatter plot of the data. (e) Use each of the regression equations in part (d) to estimate the maimum possible value of the area of the rectangle. 9. Cubic Polnomial Functions Let f a 3 b c d, a 0. (a) Show that f has either 0 or local etrema. (b) Give an eample of each possibilit in part (a). 30. Shipping Packages The U.S. Postal Service will accept a bo for domestic shipment onl if the sum of its length and girth (distance around), as shown in the figure, does not eceed 08 in. What dimensions will give a bo with a square end the largest possible volume? 3. Constructing Clinders Compare the answers to the following two construction problems. (a) A rectangular sheet of perimeter 36 cm and dimensions cm b cm is to be rolled into a clinder as shown in part (a) of the figure. What values of and give the largest volume? (b) The same sheet is to be revolved about one of the sides of length to sweep out the clinder as shown in part (b) of the figure. What values of and give the largest volume? Length Girth Distance around here Square end a b Circumference (a) Find where the absolute maimum of f occurs. (b) Let a 0 and b 0 be given as shown in the figure. Complete the following table where A is the area of the rectangle in the figure. a b A (a) 3. Constructing Cones A right triangle whose hpotenuse is 3 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this wa. h 3 (b) r

44 Section 4.4 Modeling and Optimization Finding Parameter Values What value of a makes f a have (a) a local minimum at? (b) a point of inflection at? 34. Finding Parameter Values Show that f a cannot have a local maimum for an value of a. 35. Finding Parameter Values What values of a and b make f 3 a b have (a) a local maimum at and a local minimum at 3? (b) a local minimum at 4 and a point of inflection at? 36. Inscribing a Cone Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3. (c) Writing to Learn On the same screen, graph S as a function of the beam s depth d, again taking k. Compare the graphs with one another and with our answer in part (a). What would be the effect of changing to some other value of k? Tr it. 39. Frictionless Cart A small frictionless cart, attached to the wall b a spring, is pulled 0 cm from its rest position and released at time t 0 to roll back and forth for 4 sec. Its position at time t is s 0 cos pt. (a) What is the cart s maimum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? (b) Where is the cart when the magnitude of the acceleration is greatest? What is the cart s speed then? s 37. Strength of a Beam The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. (a) Find the dimensions of the strongest beam that can be cut from a -in. diameter clindrical log. (b) Writing to Learn Graph S as a function of the beam s width w, assuming the proportionalit constant to be k. Reconcile what ou see with our answer in part (a). (c) Writing to Learn On the same screen, graph S as a function of the beam s depth d, again taking k. Compare the graphs with one another and with our answer in part (a). What would be the effect of changing to some other value of k? Tr it. 40. Electrical Current Suppose that at an time t (sec) the current i (amp) in an alternating current circuit is i cos t sin t. What is the peak (largest magnitude) current for this circuit? 4. Calculus and Geometr How close does the curve come to the point 3, 0? [Hint: If ou minimize the square of the distance, ou can avoid square roots.] (, ) = " d 0 3, 0 w 38. Stiffness of a Beam The stiffness S of a rectangular beam is proportional to its width times the cube of its depth. (a) Find the dimensions of the stiffest beam that can be cut from a -in. diameter clindrical log. (b) Writing to Learn Graph S as a function of the beam s width w, assuming the proportionalit constant to be k. Reconcile what ou see with our answer in part (a). 4. Calculus and Geometr How close does the semicircle 6 come to the point, 3? 43. Writing to Learn Is the function f ever negative? Eplain. 44. Writing to Learn You have been asked to determine whether the function f 3 4 cos cos is ever negative. (a) Eplain wh ou need to consider values of onl in the interval 0, p. (b) Is f ever negative? Eplain.

45 30 Chapter 4 Applications of Derivatives 45. Vertical Motion Two masses hanging side b side from springs have positions s sin t and s sin t, respectivel, with s and s in meters and t in seconds. D R C (a) At what times in the interval t 0 do the masses pass each other? [Hint: sin t sin t cos t.] (b) When in the interval 0 t p is the vertical distance between the masses the greatest? What is this distance? (Hint: cos t cos t.) 46. Motion on a Line The positions of two particles on the s-ais are s sin t and s sin t p 3, with s and s in meters and t in seconds. (a) At what time(s) in the interval 0 t p do the particles meet? (b) What is the farthest apart that the particles ever get? (c) When in the interval 0 t p is the distance between the particles changing the fastest? 47. Finding an Angle The trough in the figure is to be made to the dimensions shown. Onl the angle u can be varied. What value of u will maimize the trough s volume? ' m ' ' s 0 s s 48. Group Activit Paper Folding A rectangular sheet of 8 - b -in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Tr it with paper. (a) Show that L (b) What value of minimizes L? (c) What is the minimum value of L? m 0' L A 49. Sensitivit to Medicine (continuation of Eercise 48, Section 3.3) Find the amount of medicine to which the bod is most sensitive b finding the value of M that maimizes the derivative dr dm. 50. Selling Backpacks It costs ou c dollars each to manufacture and distribute backpacks. If the backpacks sell at dollars each, the number sold is given b a n b 00, c where a and b are certain positive constants. What selling price will bring a maimum profit? Standardized Test Questions L Crease P You ma use a graphing calculator to solve the following problems. 5. True or False A continuous function on a closed interval must attain a maimum value on that interval. Justif our answer. 5. True or False If f (c) 0 and f (c) is not a local maimum, then f (c) is a local minimum. Justif our answer. 53. Multiple Choice Two positive numbers have a sum of 60. What is the maimum product of one number times the square of the second number? (A) 348 (B) 3600 (C) 7,000 (D) 3,000 (E) 36, Multiple Choice A continuous function f has domain [, 5] and range [3, 30]. If f () 0 for all between and 5, what is f (5)? (A) (B) 3 (C) 5 (D) 30 (E) impossible to determine from the information given B Q (originall at A)

46 Section 4.4 Modeling and Optimization Multiple Choice What is the maimum area of a right triangle with hpotenuse 0? (A) 4 (B) 5 (C) 5 (D) 48 (E) Multiple Choice A rectangle is inscribed between the parabolas 4 and 30 as shown below: In some cases it is reasonable to assume that the rate v d dt of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, v ma be considered to be a function of alone, and v k a ka k, where the amount of product, a the amount of substance at the beginning, k a positive constant. At what value of does the rate v have a maimum? What is the maimum value of v? What is the maimum area of such a rectangle? (A) 0 (B) 40 (C) 30 (D) 50 (E) 40 Eplorations [ 3, 3] b [, 40] 57. Fermat s Principle in Optics Fermat s principle in optics states that light alwas travels from one point to another along a path that minimizes the travel time. Light from a source A is reflected b a plane mirror to a receiver at point B, as shown in the figure. Show that for the light to obe Fermat s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purel geometric argument, which ou ma prefer.) Light source A Angle of incidence Normal Plane mirror Angle of reflection Light receiver 58. Tin Pest When metallic tin is kept below 3. C, it slowl becomes brittle and crumbles to a gra powder. Tin objects eventuall crumble to this gra powder spontaneousl if kept in a cold climate for ears. The Europeans who saw tin organ pipes in their churches crumble awa ears ago called the change tin pest because it seemed to be contagious. And indeed it was, for the gra powder is a catalst for its own formation. A catalst for a chemical reaction is a substance that controls the rate of reaction without undergoing an permanent change in itself. An autocataltic reaction is one whose product is a catalst for its own formation. Such a reaction ma proceed slowl at first if the amount of catalst present is small and slowl again at the end, when most of the original substance is used up. But in between, when both the substance and its catalst product are abundant, the reaction proceeds at a faster pace. B 59. How We Cough When we cough, the trachea (windpipe) contracts to increase the velocit of the air going out. This raises the question of how much it should contract to maimize the velocit and whether it reall contracts that much when we cough. Under reasonable assumptions about the elasticit of the tracheal wall and about how the air near the wall is slowed b friction, the average flow velocit v (in cm sec) can be modeled b the equation v c r 0 r r, r 0 r r 0, where r 0 is the rest radius of the trachea in cm and c is a positive constant whose value depends in part on the length of the trachea. (a) Show that v is greatest when r 3 r 0, that is, when the trachea is about 33% contracted. The remarkable fact is that X-ra photographs confirm that the trachea contracts about this much during a cough. (b) Take r 0 to be 0.5 and c to be, and graph v over the interval 0 r 0.5. Compare what ou see to the claim that v is a maimum when r 3 r Wilson Lot Size Formula One of the formulas for inventor management sas that the average weekl cost of ordering, paing for, and holding merchandise is A q k m hq cm, q where q is the quantit ou order when things run low (shoes, radios, brooms, or whatever the item might be), k is the cost of placing an order (the same, no matter how often ou order), c is the cost of one item (a constant), m is the number of items sold each week (a constant), and h is the weekl holding cost per item (a constant that takes into account things such as space, utilities, insurance, and securit). (a) Your job, as the inventor manager for our store, is to find the quantit that will minimize A q. What is it? (The formula ou get for the answer is called the Wilson lot size formula.) (b) Shipping costs sometimes depend on order size. When the do, it is more realistic to replace k b k bq, the sum of k and a constant multiple of q. What is the most economical quantit to order now?

47 3 Chapter 4 Applications of Derivatives 6. Production Level Show that if r 6 and c are our revenue and cost functions, then the best ou can do is break even (have revenue equal cost). 6. Production Level Suppose c 3 0 0,000 is the cost of manufacturing items. Find a production level that will minimize the average cost of making items. Etending the Ideas 63. Airplane Landing Path An airplane is fling at altitude H when it begins its descent to an airport runwa that is at horizontal ground distance L from the airplane, as shown in the figure. Assume that the landing path of the airplane is the graph of a cubic polnomial function a 3 b c d where L H and 0 0. (a) What is d d at 0? (b) What is d d at L? (c) Use the values for d d at 0 and L together with 0 0 and L H to show that H[ ( L )3 3( L )]. Landing path H Cruising altitude L Airport In Eercises 64 and 65, ou might find it helpful to use a CAS. 64. Generalized Cone Problem A cone of height h and radius r is constructed from a flat, circular disk of radius a in. as described in Eploration. (a) Find a formula for the volume V of the cone in terms of and a. (b) Find r and h in the cone of maimum volume for a 4, 5, 6, 8. (c) Writing to Learn Find a simple relationship between r and h that is independent of a for the cone of maimum volume. Eplain how ou arrived at our relationship. 65. Circumscribing an Ellipse Let P, a and Q, a be two points on the upper half of the ellipse centered at 0, 5. A triangle RST is formed b using the tangent lines to the ellipse at Q and P as shown in the figure. Q(, a) S R ] (a) Show that the area of the triangle is A f [ f f 5 P(, a) where f is the function representing the upper half of the ellipse. (b) What is the domain of A? Draw the graph of A. How are the asmptotes of the graph related to the problem situation? (c) Determine the height of the triangle with minimum area. How is it related to the -coordinate of the center of the ellipse? (d) Repeat parts (a) (c) for the ellipse C B B centered at 0, B. Show that the triangle has minimum area when its height is 3B., T

48 Section 4.5 Linearization and Newton s Method 33 What ou ll learn about Linear Approimation Newton s Method Differentials Estimating Change with Differentials 4.5 Linearization and Newton s Method Linear Approimation In our stud of the derivative we have frequentl referred to the tangent line to the curve at a point. What makes that tangent line so important mathematicall is that it provides a useful representation of the curve itself if we sta close enough to the point of tangenc. We sa that differentiable curves are alwas locall linear, a fact that can best be appreciated graphicall b zooming in at a point on the curve, as Eploration shows. Absolute, Relative, and Percentage Change Sensitivit to Change... and wh Engineering and science depend on approimations in most practical applications; it is important to understand how approimation techniques work. 0 (a, f(a)) a f() Figure 4.44 The tangent to the curve f at a is the line f a f a a. Slope f'(a) EXPLORATION Appreciating Local Linearit The function f () ( 0.000) /4 0.9 is differentiable at 0 and hence locall linear there. Let us eplore the significance of this fact with the help of a graphing calculator.. Graph f () in the ZoomDecimal window. What appears to be the behavior of the function at the point (0, )?. Show algebraicall that f is differentiable at 0. What is the equation of the tangent line at (0, )? 3. Now zoom in repeatedl, keeping the cursor at (0, ). What is the long-range outcome of repeated zooming? 4. The graph of f() eventuall looks like the graph of a line. What line is it? We hope that this eploration gives ou a new appreciation for the tangent line. As ou zoom in on a differentiable function, its graph at that point actuall seems to become the graph of the tangent line! This observation that even the most complicated differentiable curve behaves locall like the simplest graph of all, a straight line is the basis for most of the applications of differential calculus. It is what allows us, for eample, to refer to the derivative as the slope of the curve or as the velocit at time t 0. Algebraicall, the principle of local linearit means that the equation of the tangent line defines a function that can be used to approimate a differentiable function near the point of tangenc. In recognition of this fact, we give the equation of the tangent line a new name: the linearization of f at a. Recall that the tangent line at (a, f (a)) has point-slope equation f (a) f ()( a) (Figure 4.44). DEFINITION Linearization If f is differentiable at a, then the equation of the tangent line, L() f (a) f (a)( a), defines the linearization of f at a. The approimation f () L() is the standard linear approimation of f at a. The point a is the center of the approimation.

49 34 Chapter 4 Applications of Derivatives EXAMPLE Finding a Linearization Find the linearization of f () at 0, and use it to approimate.0 without a calculator. Then use a calculator to determine the accurac of the approimation. Since f (0), the point of tangenc is (0, ). Since f () ( ) /, the slope of the tangent line is f (0). Thus Figure 4.45 The graph of f and its linearization at 0 and 3. (Eample ) To approimate.0, we use 0.0: L() ( 0). (Figure 4.45).0 f (0.0) L(0.0) The calculator gives , so the approimation error is We report that the error is less than 0 4. Now tr Eercise. Wh not just use a calculator? We readil admit that linearization will never replace a calculator when it comes to finding square roots. Indeed, historicall it was the other wa around. Understanding linearization, however, brings ou one step closer to understanding how the calculator finds those square roots so easil. You will get man steps closer when ou stud Talor polnomials in Chapter 9. (A linearization is just a Talor polnomial of degree.) Look at how accurate the approimation is for values of near 0. Approimation True Value Approimation As we move awa from zero (the center of the approimation), we lose accurac and the approimation becomes less useful. For eample, using L() as an approimation for f () 3 is not even accurate to one decimal place. We could do slightl better using L() to approimate f () if we were to use 3 as the center of our approimation (Figure 4.45). EXAMPLE Finding a Linearization Find the linearization of f () cos at p/ and use it to approimate cos.75 without a calculator. Then use a calculator to determine the accurac of the approimation. Since f (p/) cos (p/) 0, the point of tangenc is (p/, 0). The slope of the tangent line is f (p/) sin (p/). Thus L() 0 ( ) p p. (Figure 4.46) 0 cos Figure 4.46 The graph of f cos and its linearization at p. Near p, cos p. (Eample ) To approimate cos (.75), we use.75: cos.75 f (.75) L(.75).75 p The calculator gives cos , so the approimation error is (.75 p/) We report that the error is less than 0 3. Now tr Eercise 5.

50 Section 4.5 Linearization and Newton s Method 35 EXAMPLE 3 Approimating Binomial Powers Eample introduces a special case of a general linearization formula that applies to powers of for small values of : ( ) k k. If k is a positive integer this follows from the Binomial Theorem, but the formula actuall holds for all real values of k. (We leave the justification to ou as Eercise 7.) Use this formula to find polnomials that will approimate the following functions for values of close to zero: (a) 3 (b) (c) 5 4 (d) We change each epression to the form ( ) k, where k is a real number and is a function of that is close to 0 when is close to zero. The approimation is then given b k. (a) 3 ( ( )) /3 3 ( ) 3 (b) ( ( )) ( )( ) (c) 5 4 (( 5 4 )) / (54 ) 5 4 (d) (( ( )) / ( ) Now tr Eercise 9. EXAMPLE 4 Approimating Roots Use linearizations to approimate (a) 3 and (b) 3 3 Part of the analsis is to decide where to center the approimations. (a) Let f (). The closest perfect square to 3 is, so we center the linearization at. The tangent line at (, ) has slope f () () /. So L() (3 ).0 9. (b) Let f () 3. The closest perfect cube to 3 is 5, so we center the linearization at 5. The tangent line at (5, 5) has slope f (5) 3 (5) /3 3 ( 3 ) So 3 3 L(3) 5 (3 5) A calculator shows both approimations to be within 0 3 of the actual values. Now tr Eercise. Newton s Method Newton s method is a numerical technique for approimating a zero of a function with zeros of its linearizations. Under favorable circumstances, the zeros of the linearizations.

51 36 Chapter 4 Applications of Derivatives 0 Root sought ( 3, f( 3 )) 4 3 Fourth Third (, f( )) APPROXIMATIONS (, f( )) Second f() First converge rapidl to an accurate approimation. Man calculators use the method because it applies to a wide range of functions and usuall gets results in onl a few steps. Here is how it works. To find a solution of an equation f 0, we begin with an initial estimate, found either b looking at a graph or simpl guessing. Then we use the tangent to the curve f at, f to approimate the curve (Figure 4.47). The point where the tangent crosses the -ais is the net approimation. The number is usuall a better approimation to the solution than is. The point where the tangent to the curve at, f crosses the -ais is the net approimation 3. We continue on, using each approimation to generate the net, until we are close enough to the zero to stop. There is a formula for finding the n st approimation n from the nth approimation n. The point-slope equation for the tangent to the curve at n, f n is f n f n n. We can find where it crosses the -ais b setting 0 (Figure 4.48). 0 f n f n n Figure 4.47 Usuall the approimations rapidl approach an actual zero of f. 0 f() Point: ( n, f( n )) Slope: f'( n ) Equation: f( n ) f'( n )( n ) Root sought ( n, f( n )) f( n ) n+ n f'( n ) Figure 4.48 From n we go up to the curve and follow the tangent line down to find n. n Tangent line (graph of linearization of f at n ) f n f n f n n f n f n n f n f n n If f f n n 0 This value of is the net approimation n. Here is a summar of Newton s method. Procedure for Newton s Method. Guess a first approimation to a solution of the equation f 0. A graph of f ma help.. Use the first approimation to get a second, the second to get a third, and so on, using the formula f n n n. f EXAMPLE 5 Using Newton s Method Use Newton s method to solve Let f 3 3, then f 3 3 and f n n n f n n n 3 3n 3. n 3 The graph of f in Figure 4.49 on the net page suggests that 0.3 is a good first approimation to the zero of f in the interval 0. Then, 0.3, , , The n for n 5 all appear to equal 4 on the calculator we used for our computations. We conclude that the solution to the equation is about Now tr Eercise 5. n

52 Section 4.5 Linearization and Newton s Method 37-3 X X Y / Y Figure 4.50 A graphing calculator does the computations for Newton s method. (Eploration ) X O Figure 4.5 r f() The graph of the function r, r f { r, r. If r h, then r h. Successive approimations go back and forth between these two values, and Newton s method fails to converge. f() [ 5, 5] b [ 5, 5] Figure 4.49 A calculator graph of 3 3 suggests that 0.3 is a good first guess at the zero to begin Newton s method. (Eample 5) EXPLORATION Using Newton s Method on Your Calculator Here is an eas wa to get our calculator to perform the calculations in Newton s method. Tr it with the function f 3 3 from Eample 5.. Enter the function in Y and its derivative in Y.. On the home screen, store the initial guess into. For eample, using the initial guess in Eample 5, ou would tpe.3 X. 3. Tpe X Y/Y X and press the ENTER ke over and over. Watch as the numbers converge to the zero of f. When the values stop changing, it means that our calculator has found the zero to the etent of its displaed digits (Figure 4.50). 4. Eperiment with different initial guesses and repeat Steps and Eperiment with different functions and repeat Steps through 3. Compare each final value ou find with the value given b our calculator s built-in zerofinding feature. Newton s method does not work if f 0. In that case, choose a new starting point. Newton s method does not alwas converge. For instance (see Figure 4.5), successive approimations r h and r h can go back and forth between these two values, and no amount of iteration will bring us an closer to the zero r. If Newton s method does converge, it converges to a zero of f. However, the method ma converge to a zero that is different from the epected one if the starting value is not close enough to the zero sought. Figure 4.5 shows how this might happen. Root found 3 Root sought Starting point Figure 4.5 Newton s method ma miss the zero ou want if ou start too far awa. Differentials Leibniz used the notation d d to represent the derivative of with respect to. The notation looks like a quotient of real numbers, but it is reall a limit of quotients in which both numerator and denominator go to zero (without actuall equaling zero). That makes it trick to define d and d as separate entities. (See the margin note, Leibniz and His Notation. ) Since we reall onl need to define d and d as formal variables, we define them in terms of each other so that their quotient must be the derivative.

53 38 Chapter 4 Applications of Derivatives Leibniz and His Notation Although Leibniz did most of his calculus using d and d as separable entities, he never quite settled the issue of what the were. To him, the were infinitesimals nonzero numbers, but infinitesimall small. There was much debate about whether such things could eist in mathematics, but luckil for the earl development of calculus it did not matter: thanks to the Chain Rule, d/d behaved like a quotient whether it was one or not. Fan Chung Graham ( ) Don t be intimidated! is Dr. Fan Chung Graham s advice to oung women considering careers in mathematics. Fan Chung Graham came to the U.S. from Taiwan to earn a Ph.D. in Mathematics from the Universit of Pennslvania. She worked in the field of combinatorics at Bell Labs and Bellcore, and then, in 994, returned to her alma mater as a Professor of Mathematics. Her research interests include spectral graph theor, discrete geometr, algorithms, and communication networks. DEFINITION EXAMPLE 6 Finding the Differential d Find the differential d and evaluate d for the given values of and d. (a) 5 37,, d 0.0 (b) sin 3, p, d 0.0 (c),, d 0.05 (a) d (5 4 37) d. When and d 0.0, d (5 37)(0.0) 0.4. (b) d (3 cos 3) d. When p and d 0.0, d (3 cos 3p)( 0.0) (c) We could solve eplicitl for before differentiating, but it is easier to use implicit differentiation: d( ) d() d d d d d( ) ( )d d ( ) d Sum and Product Rules in differential form When in the original equation,, so is also. Therefore d ( )(0. 05) Now tr Eercise 9. ( ) If d 0, then the quotient of the differential d b the differential d is equal to the derivative f because We sometimes write Differentials Let f be a differentiable function. The differential d is an independent variable. The differential d is d f d. Unlike the independent variable d, the variable d is alwas a dependent variable. It depends on both and d. d f d f. d d df f d in place of d f d, calling df the differential of f. For instance, if f 3 6, then df d 3 6 6d. Ever differentiation formula like d u d v d u d d v or d s in d d u cos u du d has a corresponding differential form like d u v du dv or d sin u cos u du.

54 Section 4.5 Linearization and Newton s Method 39 EXAMPLE 7 Finding Differentials of Functions (a) d tan sec d sec d (b) d( ) d d d d d d Now tr Eercise 7. Estimating Change with Differentials Suppose we know the value of a differentiable function f at a point a and we want to predict how much this value will change if we move to a nearb point a d. If d is small, f and its linearization L at a will change b nearl the same amount (Figure 4.53). Since the values of L are simple to calculate, calculating the change in L offers a practical wa to estimate the change in f. f() f f(a d) f(a ) (a, f(a)) Tangent line d 0 a a d L f '(a)d When d is a small change in, the corresponding change in the linearization is precisel df. Figure 4.53 Approimating the change in the function f b the change in the linearization of f. In the notation of Figure 4.53, the change in f is The corresponding change in L is f f a d f a. L L a d L a f a f a a d a f a f a d. L a d Thus, the differential df f d has a geometric interpretation: The value of df at a is L, the change in the linearization of f corresponding to the change d. L a Differential Estimate of Change Let f be differentiable at a. The approimate change in the value of f when changes from a to a d is df f a d.

55 40 Chapter 4 Applications of Derivatives dr 0. a 0 A da a dr EXAMPLE 8 Estimating Change With Differentials The radius r of a circle increases from a 0 m to 0. m (Figure 4.54). Use da to estimate the increase in the circle s area A. Compare this estimate with the true change A, and find the approimation error. Since A pr, the estimated increase is The true change is da A a dr pa dr p 0 0. p m. A p 0. p p.0p m. The approimation error is A da.0p p 0.0p m. Now tr Eercise 3. Figure 4.54 When dr is small compared with a, as it is when dr 0. and a 0, the differential da pa dr gives a good estimate of A. (Eample 8) Absolute, Relative, and Percentage Change As we move from a to a nearb point a d, we can describe the change in f in three was: True Estimated Absolute change f f a d f a df f a d Relative change f df f a f a Percentage change f df f a f a Wh It s Eas to Estimate Change in Perimeter Note that the true change in Eample 9 is P (3) P () 6p 4p p, so the differential estimate in this case is perfectl accurate! Wh? Since P pr is a linear function of r, the linearization of P is the same as P itself. It is useful to keep in mind that local linearit is what makes estimation b differentials work. EXAMPLE 9 Changing Tires Inflating a biccle tire changes its radius from inches to 3 inches. Use differentials to estimate the absolute change, the relative change, and the percentage change in the perimeter of the tire. Perimeter P pr, so P dp pdr p() p 6.8. The absolute change is approimatel 6.3 inches. The relative change (when P() 4p) is approimatel p/4p The percentage change is approimatel 8 percent. Now tr Eercise 35. Another wa to interpret the change in f () resulting from a change in is the effect that an error in estimating has on the estimation of f (). We illustrate this in Eample 0. EXAMPLE 0 Estimating the Earth s Surface Area Suppose the earth were a perfect sphere and we determined its radius to be miles. What effect would the tolerance of 0. mi have on our estimate of the earth s surface area? continued

56 Section 4.5 Linearization and Newton s Method 4 The surface area of a sphere of radius r is S 4pr. The uncertaint in the calculation of S that arises from measuring r with a tolerance of dr miles is ds 8prdr. With r 3959 and dr 0., our estimate of S could be off b as much as ds 8p mi, to the nearest square mile, which is about the area of the state of Marland. Now tr Eercise 4. Angiograph An opaque de is injected into a partiall blocked arter to make the inside visible under X-ras. This reveals the location and severit of the blockage. Angioplast Opaque de Blockage A balloon-tipped catheter is inflated inside the arter to widen it at the blockage site. Inflatable balloon on catheter EXAMPLE Determining Tolerance About how accuratel should we measure the radius r of a sphere to calculate the surface area S 4pr within % of its true value? We want an inaccurac in our measurement to be small enough to make the corresponding increment S in the surface area satisf the inequalit S 00 S 4 pr. 00 We replace S in this inequalit b its approimation ds ( d S d ) dr 8prdr. r This gives 8prdr 4 pr, or dr 4 pr 00 8p r 0 0 r r. 0 We should measure r with an error dr that is no more than 0.5% of the true value. Now tr Eercise 49. EXAMPLE Unclogging Arteries In the late 830s, the French phsiologist Jean Poiseuille ( pwa-zoy ) discovered the formula we use toda to predict how much the radius of a partiall clogged arter has to be epanded to restore normal flow. His formula, V kr 4, sas that the volume V of fluid flowing through a small pipe or tube in a unit of time at a fied pressure is a constant times the fourth power of the tube s radius r. How will a 0% increase in r affect V? The differentials of r and V are related b the equation dv d V dr 4kr dr 3 dr. The relative change in V is d V 4kr 3 V kr dr 4 4 d r r. The relative change in V is 4 times the relative change in r, so a 0% increase in r will produce a 40% increase in the flow. Now tr Eercise 5.

57 4 Chapter 4 Applications of Derivatives Sensitivit to Change The equation df f d tells how sensitive the output of f is to a change in input at different values of. The larger the value of f at, the greater the effect of a given change d. EXAMPLE 3 Finding Depth of a Well You want to calculate the depth of a well from the equation s 6t b timing how long it takes a heav stone ou drop to splash into the water below. How sensitive will our calculations be to a 0. sec error in measuring the time? The size of ds in the equation ds 3tdt depends on how big t is. If t sec, the error caused b dt 0. is onl ds ft. Three seconds later at t 5 sec, the error caused b the same dt is ds ft. Now tr Eercise 53. Quick Review 4.5 (For help, go to Sections 3.3, 3.6, and 3.9.) In Eercises and, find d d.. sin. cos In Eercises 3 and 4, solve the equation graphicall. 3. e In Eercises 5 and 6, let f e. Write an equation for the line tangent to f at c. 5. c 0 6. c 7. Find where the tangent line in (a) Eercise 5 and (b) Eercise 6 crosses the -ais. 8. Let g be the function whose graph is the tangent line to the graph of f 3 4 at. Complete the table. In Eercises 9 and 0, graph f and its tangent line at c. 9. c.5, f sin 0. f g , 3 c 4, f { 3, 3 Section 4.5 Eercises In Eercises 6, (a) find the linearization L of f at a. (b) How accurate is the approimation L a 0. f a 0.? See the comparisons following Eample.. f 3 3, a. f 9, a 4 3. f, a 4. f ln, a 0 5. f tan, a p 6. f cos, a 0 7. Show that the linearization of f k at 0 is L k. 8. Use the linearization k k to approimate the following. State how accurate our approimation is. (a) (.00) 00 (b) In Eercises 9 and 0, use the linear approimation k k to find an approimation for the function f for values of near zero. 9. (a) f 6 (b) f (c) f

58 Section 4.5 Linearization and Newton s Method (a) f 4 3 /3 (b) f (c) f 3 ( ) In Eercises 4, approimate the root b using a linearization centered at an appropriate nearb number In Eercises 5 8, use Newton s method to estimate all real solutions of the equation. Make our answers accurate to 6 decimal places sin In Eercises 9 6, (a) find d, and (b) evaluate d for the given value of and d ,, d ,, d 0.. ln,, d 0.0., 0, d e sin, p, d csc ( 3 ),, d , 0, d ,, d 0.05 In Eercises 7 30, find the differential. 7. d( ) 8. d(e 5 5 ) 9. d(arctan 4) 30. d(8 8 ) In Eercises 3 34, the function f changes value when changes from a to a d. Find (a) the true change f f a d f a. (b) the estimated change df f a d. (c) the approimation error f df. f() 33. f, a 0.5, d f 4, a, d 0.0 In Eercises 35 40, write a differential formula that estimates the given change in volume or surface area. Then use the formula to estimate the change when the dependent variable changes from 0 cm to 0.05 cm. 35. Volume The change in the volume V 4 3 pr 3 of a sphere when the radius changes from a to a dr 36. Surface Area The change in the surface area S 4pr of a sphere when the radius changes from a to a dr V r 4 3, S 4 r Volume The change in the volume V 3 of a cube when the edge lengths change from a to a d 38. Surface Area The change in the surface area S 6 of a cube when the edge lengths change from a to a d 39. Volume The change in the volume V pr h of a right circular clinder when the radius changes from a to a dr and the height does not change 40. Surface Area The change in the lateral surface area S prh of a right circular clinder when the height changes from a to a dh and the radius does not change V r h, r r S rh h V 3, S 6 ( a, f( a)) df f'(a)d d Tangent O a a d 3. f, a 0, d f 3, a, d 0. f f( a d) f( a) In Eercises 4 44, use differentials to estimate the maimum error in measurement resulting from the tolerance of error in the dependent variable. Epress answers to the nearest tenth, since that is the precision used to epress the tolerance. 4. The area of a circle with radius 0 0. in. 4. The volume of a sphere with radius in. 43. The volume of a cube with side 5 0. cm.

59 44 Chapter 4 Applications of Derivatives 44. The area of an equilateral triangle with side cm. 45. Linear Approimation Let f be a function with f 0 and f cos. (a) Find the linearization of f at 0. (b) Estimate the value of f at 0.. (c) Writing to Learn Do ou think the actual value of f at 0. is greater than or less than the estimate in part (b)? Eplain. 46. Epanding Circle The radius of a circle is increased from.00 to.0 m. (a) Estimate the resulting change in area. (b) Estimate as a percentage of the circle s original area. 47. Growing Tree The diameter of a tree was 0 in. During the following ear, the circumference increased in. About how much did the tree s diameter increase? the tree s cross section area? 48. Percentage Error The edge of a cube is measured as 0 cm with an error of %. The cube s volume is to be calculated from this measurement. Estimate the percentage error in the volume calculation. 49. Tolerance About how accuratel should ou measure the side of a square to be sure of calculating the area to within % of its true value? 50. Tolerance (a) About how accuratel must the interior diameter of a 0-m high clindrical storage tank be measured to calculate the tank s volume to within % of its true value? (b) About how accuratel must the tank s eterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount? 5. Minting Coins A manufacturer contracts to mint coins for the federal government. The coins must weigh within 0.% of their ideal weight, so the volume must be within 0.% of the ideal volume. Assuming the thickness of the coins does not change, what is the percentage change in the volume of the coin that would result from a 0.% increase in the radius? 5. Tolerance The height and radius of a right circular clinder are equal, so the clinder s volume is V ph 3. The volume is to be calculated with an error of no more than % of the true value. Find approimatel the greatest error that can be tolerated in the measurement of h, epressed as a percentage of h. 53. Estimating Volume You can estimate the volume of a sphere b measuring its circumference with a tape measure, dividing b p to get the radius, then using the radius in the volume formula. Find how sensitive our volume estimate is to a /8 in. error in the circumference measurement b filling in the table below for spheres of the given sizes. Use differentials when filling in the last column. Sphere Tpe True Radius Tape Error Radius Error Volume Error Orange in. /8 in. Melon 4 in. /8 in. Beach Ball 7 in. /8 in. 54. Estimating Surface Area Change the heading in the last column of the table in Eercise 53 to Surface Area Error and find how sensitive the measure of surface area is to a /8 in. error in estimating the circumference of the sphere. 55. The Effect of Flight Maneuvers on the Heart The amount of work done b the heart s main pumping chamber, the left ventricle, is given b the equation W PV V dv, g where W is the work per unit time, P is the average blood pressure, V is the volume of blood pumped out during the unit of time, d ( delta ) is the densit of the blood, v is the average velocit of the eiting blood, and g is the acceleration of gravit. When P, V, d, and v remain constant, W becomes a function of g, and the equation takes the simplified form W a b a, b constant. g As a member of NASA s medical team, ou want to know how sensitive W is to apparent changes in g caused b flight maneuvers, and this depends on the initial value of g. As part of our investigation, ou decide to compare the effect on W of a given change dg on the moon, where g 5. ft sec, with the effect the same change dg would have on Earth, where g 3 ft sec. Use the simplified equation above to find the ratio of dw moon to dw Earth. 56. Measuring Acceleration of Gravit When the length L of a clock pendulum is held constant b controlling its temperature, the pendulum s period T depends on the acceleration of gravit g. The period will therefore var slightl as the clock is moved from place to place on the earth s surface, depending on the change in g. B keeping track of T, we can estimate the variation in g from the equation T p L g that relates T, g, and L. (a) With L held constant and g as the independent variable, calculate dt and use it to answer parts (b) and (c). (b) Writing to Learn If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Eplain. (c) A clock with a 00-cm pendulum is moved from a location where g 980 cm sec to a new location. This increases the period b dt 0.00 sec. Find dg and estimate the value of g at the new location.

60 Section 4.5 Linearization and Newton s Method 45 Standardized Test Questions You ma use a graphing calculator to solve the following problems. 57. True or False Newton s method will not find the zero of f () ( ) if the first guess is greater than. Justif our answer. 58. True or False If u and v are differentiable functions, then d(uv) du dv. Justif our answer. 59. Multiple Choice What is the linearization of f () e at? (A) e (B) e (C) e (D) e (E) e( ) 60. Multiple Choice If tan, p, and d 0.5, what does d equal? (A) 0.5 (B) 0.5 (C) 0 (D) 0.5 (E) Multiple Choice If Newton s method is used to find the zero of f () 3, what is the third estimate if the first estimate is? (A) 3 4 (B) 3 (C) 8 5 (D) 8 (E) 3 6. Multiple Choice If the linearization of 3 at 64 is used to approimate 3 66, what is the percentage error? (A) 0.0% (B) 0.04% (C) 0.4% (D) % (E) 4% Eplorations 63. Newton s Method Suppose our first guess in using Newton s method is luck in the sense that is a root of f 0. What happens to and later approimations? 64. Oscillation Show that if h 0, appling Newton s method to, 0 f {, 0 leads to h if h, and to h if h. Draw a picture that shows what is going on. 65. Approimations that Get Worse and Worse Appl Newton s method to f 3 with, and calculate, 3, 4, and 5. Find a formula for n. What happens to n as n? Draw a picture that shows what is going on. 66. Quadratic Approimations (a) Let Q b 0 b a b a be a quadratic approimation to f at a with the properties: iii. Q a f a, iii. Q a f a, iii. Q a f a. Determine the coefficients b 0, b, and b. (b) Find the quadratic approimation to f at 0. (c) Graph f and its quadratic approimation at 0. Then zoom in on the two graphs at the point 0,. Comment on what ou see. (d) Find the quadratic approimation to g at. Graph g and its quadratic approimation together. Comment on what ou see. (e) Find the quadratic approimation to h at 0. Graph h and its quadratic approimation together. Comment on what ou see. (f) What are the linearizations of f, g, and h at the respective points in parts (b), (d), and (e)? 67. Multiples of Pi Store an number as X in our calculator. Then enter the command X tan(x) X and press the ENTER ke repeatedl until the displaed value stops changing. The result is alwas an integral multiple of p. Wh is this so? [Hint: These are zeros of the sine function.] Etending the Ideas 68. Formulas for Differentials Verif the following formulas. (a) d c 0 (c a constant) (b) d cu c du (c a constant) (c) d u v du dv (d) d u v u dv v du (e) d( u v ) v du v udv (f) d u n nu n du 69. Linearization Show that the approimation of tan b its linearization at the origin must improve as 0 b showing that lim tan The Linearization is the Best Linear Approimation Suppose that f is differentiable at a and that g m a c (m and c constants). If the error E f g were small enough near a, we might think of using g as a linear approimation of f instead of the linearization L f a f a a. Show that if we impose on g the conditions ii. E a 0, The error is zero at a. The error is negligible when ii. lim 0, a E a compared with a. then g f a f a a. Thus, the linearization gives the onl linear approimation whose error is both zero at a and negligible in comparison with a. The linearization, L(): f(a) f'(a)( a) (a, f(a)) Some other linear approimation, g(): m( a) c f() a 7. Writing to Learn Find the linearization of f sin at 0. How is it related to the individual linearizations for and sin?

61 46 Chapter 4 Applications of Derivatives 4.6 What ou ll learn about Related Rate Equations Solution Strateg Simulating Related Motion... and wh Related rate problems are at the heart of Newtonian mechanics; it was essentiall to solve such problems that calculus was invented. Related Rates Related Rate Equations Suppose that a particle P, is moving along a curve C in the plane so that its coordinates and are differentiable functions of time t. If D is the distance from the origin to P, then using the Chain Rule we can find an equation that relates dd dt, d dt, and d dt. D ( d D dt d d dt dt An equation involving two or more variables that are differentiable functions of time t can be used to find an equation that relates their corresponding rates. EXAMPLE Finding Related Rate Equations (a) Assume that the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that relates dv/dt and dr/dt. (b) Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume of the cone. Find an equation that relates dv dt, dr dt, and dh dt. (a) V 4 pr 3 Volume formula for a sphere 3 d V 4pr dt d r dt ) (b) V p 3 r h Cone volume formula d V p dt 3 ( r d h r d r dt dt h) p 3 ( r d h rh d r dt dt ) Now tr Eercise 3. Solution Strateg What has alwas distinguished calculus from algebra is its abilit to deal with variables that change over time. Eample illustrates how eas it is to move from a formula relating static variables to a formula that relates their rates of change: simpl differentiate the formula implicitl with respect to t. This introduces an important categor of problems called related rate problems that still constitutes one of the most important applications of calculus. We introduce a strateg for solving related rate problems, similar to the strateg we introduced for ma-min problems earlier in this chapter. Strateg for Solving Related Rate Problems. Understand the problem. In particular, identif the variable whose rate of change ou seek and the variable (or variables) whose rate of change ou know.. Develop a mathematical model of the problem. Draw a picture (man of these problems involve geometric figures) and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Onl constant quantities can be assigned numerical values at the start. continued

62 Section 4.6 Related Rates Write an equation relating the variable whose rate of change ou seek with the variable(s) whose rate of change ou know. The formula is often geometric, but it could come from a scientific application. 4. Differentiate both sides of the equation implicitl with respect to time t. Be sure to follow all the differentiation rules. The Chain Rule will be especiall critical, as ou will be differentiating with respect to the parameter t. 5. Substitute values for an quantities that depend on time. Notice that it is onl safe to do this after the differentiation step. Substituting too soon freezes the picture and makes changeable variables behave like constants, with zero derivatives. 6. Interpret the solution. Translate our mathematical result into the problem setting (with appropriate units) and decide whether the result makes sense. We illustrate the strateg in Eample. Range finder Balloon 500 ft Figure 4.55 The picture shows how h and u are related geometricall. We seek dh/dt when u p/4 and du/dt 0.4 rad/min. (Eample ) Unit Analsis in Eample θ A careful analsis of the units in Eample gives dh/dt (500 ft)( ) (0.4 rad/min) 40 ft rad/min. Remember that radian measure is actuall dimensionless, adaptable to whatever unit is applied to the unit circle. The linear units in Eample are measured in feet, so ft rad is simpl ft. h EXAMPLE A Rising Balloon A hot-air balloon rising straight up from a level field is tracked b a range finder 500 feet from the lift-off point. At the moment the range finder s elevation angle is p/4, the angle is increasing at the rate of 0.4 radians per minute. How fast is the balloon rising at that moment? We will carefull identif the si steps of the strateg in this first eample. Step : Let h be the height of the balloon and let u be the elevation angle. We seek: dh/dt We know: du/dt 0.4 rad/min Step : We draw a picture (Figure 4.55). We label the horizontal distance 500 ft because it does not change over time. We label the height h and the angle of elevation u. Notice that we do not label the angle p/4, as that would freeze the picture. h Step 3: We need a formula that relates h and u. Since tan u, we get 5 00 h 500 tan u. Step 4: Differentiate implicitl: d d (h) (500 tan u) d t d t d h 500 sec u d u dt dt Step 5: Let du/dt 0.4 and let u p/4. (Note that it is now safe to specif our moment in time.) d h 500 sec dt p 4 (0.4) 500( ) (0.4) 40. Step 6: At the moment in question, the balloon is rising at the rate of 40 ft/min. Now tr Eercise. EXAMPLE 3 A Highwa Chase A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 0 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car? continued

63 48 Chapter 4 Applications of Derivatives z Figure 4.56 A sketch showing the variables in Eample 3. We know d/dt and dz/dt, and we seek d/dt. The variables,, and z are related b the Pthagorean Theorem: z. We carr out the steps of the strateg. Let be the distance of the speeding car from the intersection, let be the distance of the police cruiser from the intersection, and let z be the distance between the car and the cruiser. Distances and z are increasing, but distance is decreasing; so d/dt is negative. We seek: d/dt We know: dz/dt 0 mph and d/dt 60 mph A sketch (Figure 4.56) shows that,, and z form three sides of a right triangle. We need to relate those three variables, so we use the Pthagorean Theorem: z Differentiating implicitl with respect to t, we get d d z d z, which reduces to d d z d z. dt dt dt dt dt dt We now substitute the numerical values for,, dz/dt, d/dt, and z (which equals ): (0.8) d (0.6)( 60) (0.8) ) (0.6 (0) dt (0.8) d 36 ()(0) dt d 70 dt At the moment in question, the car s speed is 70 mph. Now tr Eercise 3. Figure 4.57 In Eample 4, the cone of water is increasing in volume inside the reservoir. We know dv/dt and we seek dh/dt. Similar triangles enable us to relate V directl to h. h 5 ft r 0 ft EXAMPLE 4 Filling a Conical Tank Water runs into a conical tank at the rate of 9 ft 3 min. The tank stands point down and has a height of 0 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? We carr out the steps of the strateg. Figure 4.57 shows a partiall filled conical tank. The tank itself does not change over time; what we are interested in is the changing cone of water inside the tank. Let V be the volume, r the radius, and h the height of the cone of water. We seek: dh/dt We know: dv/dt = 9 ft 3 /min We need to relate V and h. The volume of the cone of water is V 3 pr h, but this formula also involves the variable r, whose rate of change is not given. We need to either find dr/dt (see Solution ) or eliminate r from the equation, which we can do b using the similar triangles in Figure 4.57 to relate r and h: Therefore, r 5, or simpl r h h 0. V 3 p h p h h 3. continued

64 Section 4.6 Related Rates 49 Differentiate with respect to t: d V p 3h d h p dt dt 4 h d h. dt Let h 6 and dv/dt 9; then solve for dh/dt: 9 p 4 (6) d h dt d h 0.3 dt p At the moment in question, the water level is rising at 0.3 ft/min. The similar triangle relationship r h also implies that d r dt d h dt and that r 3 when h 6. So, we could have left all three variables in the formula V 3 r h and proceeded as follows: d V dt 3 p r d r h r d h dt dt 3 p r d d h d h t h r dt 9 3 p (3) d h d h dt (6) (3) dt 9 9p d h dt d h dt p This is obviousl more complicated than the one-variable approach. In general, it is computationall easier to simplif epressions as much as possible before ou differentiate. Now tr Eercise 7. Simulating Related Motion Parametric mode on a grapher can be used to simulate the motion of moving objects when the motion of each can be epressed as a function of time. In a classic related rate problem, the top end of a ladder slides verticall down a wall as the bottom end is pulled horizontall awa from the wall at a stead rate. Eploration shows how ou can use our grapher to simulate the related movements of the two ends of the ladder. EXPLORATION The Sliding Ladder A 0-foot ladder leans against a vertical wall. The base of the ladder is pulled awa from the wall at a constant rate of ft/sec.. Eplain wh the motion of the two ends of the ladder can be represented b the parametric equations given on the net page. continued

65 50 Chapter 4 Applications of Derivatives XT T YT 0 XT 0 YT 0 (T). What minimum and maimum values of T make sense in this problem? 3. Put our grapher in parametric and simultaneous modes. Enter the parametric equations and change the graphing stle to 0 (the little ball) if our grapher has this feature. Set Tmin 0, Tma 5, Tstep 5/0, Xmin, Xma 7, Xscl 0, Ymin, Yma, and Yscl 0. You can speed up the action b making the denominator in the Tstep smaller or slow it down b making it larger. 4. Press GRAPH and watch the two ends of the ladder move as time changes. Do both ends seem to move at a constant rate? 5. To see the simulation again, enter ClrDraw from the DRAW menu. 6. If represents the vertical height of the top of the ladder and the distance of the bottom from the wall, relate and and find d dt in terms of and. (Remember that d dt.) 7. Find d dt when t 3 and interpret its meaning. Wh is it negative? 8. In theor, how fast is the top of the ladder moving as it hits the ground? Figure 4.58 shows ou how to write a calculator program that animates the falling ladder as a line segment. PROGRAM : LADDER : For (A, 0, 5,.5) : ClrDraw : Line(,+ (00 (A) ), +A, ) : If A=0 : Pause : End WINDOW Xmin= Xma=0 Xscl=0 Ymin= Yma=3 Yscl=0 Xres= Figure 4.58 This 5-step program (with the viewing window set as shown) will animate the ladder in Eploration. Be sure an functions in the Y register are turned off. Run the program and the ladder appears against the wall; push ENTER to start the bottom moving awa from the wall. For an enhanced picture, ou can insert the commands :Pt-On(, (00 (A) ),) and :Pt-On( A,,) on either side of the middle line of the program. Quick Review 4.6 (For help, go to Sections.,.4, and 3.7.) In Eercises and, find the distance between the points A and B.. A 0, 5, B 7, 0. A 0, a, B b, 0 In Eercises 3 6, find d d sin 5. tan 6. ln In Eercises 7 and 8, find a parametrization for the line segment with endpoints A and B. 7. A,, B 4, 3 8. A 0, 4, B 5, 0 In Eercises 9 and 0, let cos t, sin t. Find a parameter interval that produces the indicated portion of the graph. 9. The portion in the second and third quadrants, including the points on the aes. 0. The portion in the fourth quadrant, including the points on the aes.

66 Section 4.6 Related Rates 5 Section 4.6 Eercises In Eercises 4, assume all variables are differentiable functions of t.. Area The radius r and area A of a circle are related b the equation A pr. Write an equation that relates da dt to dr dt.. Surface Area The radius r and surface area S of a sphere are related b the equation S 4pr. Write an equation that relates ds dt to dr dt. 3. Volume The radius r, height h, and volume V of a right circular clinder are related b the equation V pr h. (a) How is dv dt related to dh dt if r is constant? (b) How is dv dt related to dr dt if h is constant? (c) How is dv dt related to dr dt and dh dt if neither r nor h is constant? 4. Electrical Power The power P (watts) of an electric circuit is related to the circuit s resistance R (ohms) and current I (amperes) b the equation P RI. (a) How is dp dt related to dr dt and di dt? (b) How is dr dt related to di dt if P is constant? 5. Diagonals If,, and z are lengths of the edges of a rectangular bo, the common length of the bo s diagonals is s. z How is ds dt related to d dt, d dt, and dz dt? 6. Area If a and b are the lengths of two sides of a triangle, and u the measure of the included angle, the area A of the triangle is A ab sin u. How is da dt related to da dt, db dt, and du dt? 7. Changing Voltage The voltage V (volts), current I (amperes), and resistance R (ohms) of an electric circuit like the one shown here are related b the equation V IR. Suppose that V is increasing at the rate of volt sec while I is decreasing at the rate of 3 amp sec. Let t denote time in sec. 9. Changing Dimensions in a Rectangle The length of a rectangle is decreasing at the rate of cm sec while the width w is increasing at the rate of cm sec. When cm and w 5 cm, find the rates of change of (a) the area, (b) the perimeter, and (c) the length of a diagonal of the rectangle. (d) Writing to Learn Which of these quantities are decreasing, and which are increasing? Eplain. 0. Changing Dimensions in a Rectangular Bo Suppose that the edge lengths,, and z of a closed rectangular bo are changing at the following rates: d m sec, d m sec, d z m sec. dt dt dt Find the rates at which the bo s (a) volume, (b) surface area, and (c) diagonal length s z are changing at the instant when 4, 3, and z.. Inflating Balloon A spherical balloon is inflated with helium at the rate of 00p ft 3 min. (a) How fast is the balloon s radius increasing at the instant the radius is 5 ft? (b) How fast is the surface area increasing at that instant?. Growing Raindrop Suppose that a droplet of mist is a perfect sphere and that, through condensation, the droplet picks up moisture at a rate proportional to its surface area. Show that under these circumstances the droplet s radius increases at a constant rate. 3. Air Traffic Control An airplane is fling at an altitude of 7 mi and passes directl over a radar antenna as shown in the figure. When the plane is 0 mi from the antenna s 0, the radar detects that the distance s is changing at the rate of 300 mph. What is the speed of the airplane at that moment? V + I 7 mi s R (a) What is the value of dv dt? (b) What is the value of di dt? (c) Write an equation that relates dr dt to dv dt and di dt. (d) Writing to Learn Find the rate at which R is changing when V volts and I amp. Is R increasing, or decreasing? Eplain. 8. Heating a Plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.0 cm sec. At what rate is the plate s area increasing when the radius is 50 cm? 4. Fling a Kite Inge flies a kite at a height of 300 ft, the wind carring the kite horizontall awa at a rate of 5 ft sec. How fast must she let out the string when the kite is 500 ft awa from her? 5. Boring a Clinder The mechanics at Lincoln Automotive are reboring a 6-in. deep clinder to fit a new piston. The machine the are using increases the clinder s radius one-thousandth of an inch ever 3 min. How rapidl is the clinder volume increasing when the bore (diameter) is in.?

67 5 Chapter 4 Applications of Derivatives 6. Growing Sand Pile Sand falls from a conveor belt at the rate of 0 m 3 min onto the top of a conical pile. The height of the pile is alwas three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Give our answer in cm min. 7. Draining Conical Reservoir Water is flowing at the rate of 50 m 3 min from a concrete conical reservoir (verte down) of base radius 45 m and height 6 m. (a) How fast is the water level falling when the water is 5 m deep? (b) How fast is the radius of the water s surface changing at that moment? Give our answer in cm min. 8. Draining Hemispherical Reservoir Water is flowing at the rate of 6 m 3 min from a reservoir shaped like a hemispherical bowl of radius 3 m, shown here in profile. Answer the following questions given that the volume of water in a hemispherical bowl of radius R is V p 3 3R when the water is units deep. Water level Center of sphere (a) At what rate is the water level changing when the water is 8 m deep? (b) What is the radius r of the water s surface when the water is m deep? (c) At what rate is the radius r changing when the water is 8 m deep? r 3 m 0. Filling a Trough A trough is 5 ft long and 4 ft across the top as shown in the figure. Its ends are isosceles triangles with height 3 ft. Water runs into the trough at the rate of.5 ft 3 min. How fast is the water level rising when it is ft deep?. Hauling in a Dingh A dingh is pulled toward a dock b a rope from the bow through a ring on the dock 6 ft above the bow as shown in the figure. The rope is hauled in at the rate of ft sec. (a) How fast is the boat approaching the dock when 0 ft of rope are out? (b) At what rate is angle changing at that moment?. Rising Balloon A balloon is rising verticall above a level, straight road at a constant rate of ft sec. Just when the balloon is 65 ft above the ground, a biccle moving at a constant rate of 7 ft sec passes under it. How fast is the distance between the biccle and balloon increasing 3 sec later (see figure)? (t) 3 ft 5 ft 4 ft.5 ft 3 /min ft Ring at edge of dock u 6' 9. Sliding Ladder A 3-ft ladder is leaning against a house (see figure) when its base starts to slide awa. B the time the base is ft from the house, the base is moving at the rate of 5 ft sec. s(t) (t) 3-ft ladder 0 (t) 0 u (a) How fast is the top of the ladder sliding down the wall at that moment? (b) At what rate is the area of the triangle formed b the ladder, wall, and ground changing at that moment? (c) At what rate is the angle u between the ladder and the ground changing at that moment? (t) In Eercises 3 and 4, a particle is moving along the curve f. 3. Let f. 0 If d dt 3 cm sec, find d dt at the point where (a). (b) 0. (c) Let f 3 4. If d dt cm sec, find d dt at the point where (a) 3. (b). (c) 4.

68 Section 4.6 Related Rates Particle Motion A particle moves along the parabola in the first quadrant in such a wa that its -coordinate (in meters) increases at a constant rate of 0 m sec. How fast is the angle of inclination u of theline joining the particle to the origin changing when 3? 6. Particle Motion A particle moves from right to left along the parabolic curve in such a wa that its -coordinate (in meters) decreases at the rate of 8 m sec. How fast is the angle of inclination u of the line joining the particle to the origin changing when 4? 7. Melting Ice A spherical iron ball is coated with a laer of ice of uniform thickness. If the ice melts at the rate of 8 ml min, how fast is the outer surface area of ice decreasing when the outer diameter (ball plus ice) is 0 cm? 8. Particle Motion A particle P, is moving in the coordinate plane in such a wa that d dt m sec and d dt 5 m sec. How fast is the particle s distance from the origin changing as it passes through the point 5,? 9. Moving Shadow A man 6 ft tall walks at the rate of 5 ft sec toward a streetlight that is 6 ft above the ground. At what rate is the length of his shadow changing when he is 0 ft from the base of the light? 30. Moving Shadow A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft awa from the light as shown below. How fast is the ball s shadow moving along the ground sec later? (Assume the ball falls a distance s 6t in t sec.) 50-ft pole Light 0 Ball at time t 0 / sec later Shadow 30 (t) NOT TO SCALE 3. Moving Race Car You are videotaping a race from a stand 3 ft from the track, following a car that is moving at 80 mph (64 ft sec) as shown in the figure. About how fast will our camera angle u be changing when the car is right in front of ou? a half second later? Camera 3. Speed Trap A highwa patrol airplane flies 3 mi above a level, straight road at a constant rate of 0 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi the line-of-sight distance is decreasing at the rate of 60 mph. Find the car s speed along the highwa. 3 mi Plane Building s Shadow On a morning of a da when the sun will pass directl overhead, the shadow of an 80-ft building on level ground is 60 ft long as shown in the figure. At the moment in question, the angle u the sun makes with the ground is increasing at the rate of 0.7 min. At what rate is the shadow length decreasing? Epress our answer in in. min, to the nearest tenth. (Remember to use radians.) 34. Walkers A and B are walking on straight streets that meet at right angles. A approaches the intersection at m sec and B moves awa from the intersection at m sec as shown in the figure. At what rate is the angle u changing when A is 0 m from the intersection and B is 0 m from the intersection? Epress our answer in degrees per second to the nearest degree. A s(t) NOT TO SCALE 80' 60' (t) Car θ 3' O B Race Car 35. Moving Ships Two ships are steaming awa from a point O along routes that make a 0 angle. Ship A moves at 4 knots (nautical miles per hour; a nautical mile is 000 ards). Ship B moves at knots. How fast are the ships moving apart when OA 5 and OB 3 nautical miles?

69 54 Chapter 4 Applications of Derivatives Standardized Test Questions You ma use a graphing calculator to solve the following problems. 36. True or False If the radius of a circle is epanding at a constant rate, then its circumference is increasing at a constant rate. Justif our answer. 37. True or False If the radius of a circle is epanding at a constant rate, then its area is increasing at a constant rate. Justif our answer. 38. Multiple Choice If the volume of a cube is increasing at 4 in 3 / min and each edge of the cube is increasing at in./min, what is the length of each edge of the cube? (A) in. (B) in. (C) 3 in. (D) 4 in. (E) 8 in. 39. Multiple Choice If the volume of a cube is increasing at 4 in 3 / min and the surface area of the cube is increasing at in / min, what is the length of each edge of the cube? (A) in. (B) in. (C) 3 in. (D) 4 in. (E) 8 in. 40. Multiple Choice A particle is moving around the unit circle (the circle of radius centered at the origin). At the point (0.6, 0.8) the particle has horizontal velocit d/dt 3. What is its vertical velocit d/dt at that point? (A) (B) 3.75 (C).5 (D) 3.75 (E) Multiple Choice A clindrical rubber cord is stretched at a constant rate of cm per second. Assuming its volume does not change, how fast is its radius shrinking when its length is 00 cm and its radius is cm? (A) 0 cm/sec (B) 0.0 cm/sec (C) 0.0 cm/sec (D) cm/sec (E) cm/sec Eplorations 4. Making Coffee Coffee is draining from a conical filter into a clindrical coffeepot at the rate of 0 in 3 min. 43. Cost, Revenue, and Profit A compan can manufacture items at a cost of c dollars, a sales revenue of r dollars, and a profit of p r c dollars (all amounts in thousands). Find dc dt, dr dt, and dp dt for the following values of and d dt. (a) r 9, c 3 6 5, and d dt 0. when. (b) r 70, c , and d dt 0.05 when Group Activit Cardiac Output In the late 860s, Adolf Fick, a professor of phsiolog in the Facult of Medicine in Würtzberg, German, developed one of the methods we use toda for measuring how much blood our heart pumps in a minute. Your cardiac output as ou read this sentence is probabl about 7 liters a minute. At rest it is likel to be a bit under 6 L min. If ou are a trained marathon runner running a marathon, our cardiac output can be as high as 30 L min. Your cardiac output can be calculated with the formula Q D, where Q is the number of milliliters of CO ou ehale in a minute and D is the difference between the CO concentration (ml L) in the blood pumped to the lungs and the CO concentration in the blood returning from the lungs. With Q 33 ml min and D ml L, 33 ml min 5.68 L min, 4 ml L fairl close to the 6 L min that most people have at basal (resting) conditions. (Data courtes of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State Universit.) Suppose that when Q 33 and D 4, we also know that D is decreasing at the rate of units a minute but that Q remains unchanged. What is happening to the cardiac output? 6" 6" How fast is this level falling? Etending the Ideas 45. Motion along a Circle A wheel of radius ft makes 8 revolutions about its center ever second. (a) Eplain how the parametric equations cos u, sin u can be used to represent the motion of the wheel. (b) Epress u as a function of time t. (c) Find the rate of horizontal movement and the rate of vertical movement of a point on the edge of the wheel when it is at the position given b u p 4, p, and p. 6" How fast is this level rising? (a) How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? (b) How fast is the level in the cone falling at that moment? 46. Ferris Wheel A Ferris wheel with radius 30 ft makes one revolution ever 0 sec. (a) Assume that the center of the Ferris wheel is located at the point 0, 40, and write parametric equations to model its motion. [Hint: See Eercise 45.] (b) At t 0 the point P on the Ferris wheel is located at 30, 40. Find the rate of horizontal movement, and the rate of vertical movement of the point P when t 5 sec and t 8 sec.

70 Section 4.6 Related Rates Industrial Production (a) Economists often use the epression rate of growth in relative rather than absolute terms. For eample, let u f t be the number of people in the labor force at time t in a given industr. (We treat this function as though it were differentiable even though it is an integer-valued step function.) (b) Suppose that the labor force in part (a) is decreasing at the rate of % per ear while the production per person is increasing at the rate of 3% per ear. Is the total production increasing, or is it decreasing, and at what rate? Let v g t be the average production per person in the labor force at time t. The total production is then uv. If the labor force is growing at the rate of 4% per ear du dt 0.04u and the production per worker is growing at the rate of 5% per ear dv dt 0.05v, find the rate of growth of the total production,. Quick Quiz for AP* Preparation: Sections You ma use a graphing calculator to solve the following problems.. Multiple Choice If Newton's method is used to approimate the real root of 3 0, what would the third approimation, 3, be if the first approimation is? (A) (B) (C) (D) (E).977. Multiple Choice The sides of a right triangle with legs and and hpotenuse z increase in such a wa that dz dt and d dt 3 d dt. At the instant when 4 and 3, what is d dt? (A) (B) (C) 59.0 (D) (E) Free Response (a) Approimate 6 b using the linearization of at the point (5, 5). Show the computation that leads to our conclusion. (b) Approimate 6 b using a first guess of 5 and one iteration of Newton's method to approimate the zero of 6. Show the computation that leads to our conclusion. (c) Approimate 3 6 b using an appropriate linearization. Show the computation that leads to our conclusion. (A) (B) (C) (D) 5 (E) Multiple Choice An observer 70 meters south of a railroad crossing watches an eastbound train traveling at 60 meters per second. At how man meters per second is the train moving awa from the observer 4 seconds after it passes through the intersection? Chapter 4 Ke Terms absolute change (p. 40) absolute maimum value (p. 87) absolute minimum value (p. 87) antiderivative (p. 00) antidifferentiation (p. 00) arithmetic mean (p. 04) average cost (p. 4) center of linear approimation (p. 33) concave down (p. 07) concave up (p. 07) concavit test (p. 08) critical point (p. 90) decreasing function (p. 98) differential (p. 37) differential estimate of change (p. 39) differential of a function (p. 39) etrema (p. 87) Etreme Value Theorem (p. 88) first derivative test (p. 05) first derivative test for local etrema (p. 05) geometric mean (p. 04) global maimum value (p. 77) global minimum value (p. 77) increasing function (p. 98) linear approimation (p. 33) linearization (p. 33) local linearit (p. 33) local maimum value (p. 89) local minimum value (p. 89) logistic curve (p. 0) logistic regression (p. ) marginal analsis (p. 3) marginal cost and revenue (p. 3) Mean Value Theorem (p. 96) monotonic function (p. 98) Newton s method (p. 35) optimization (p. 9) percentage change (p. 40) point of inflection (p. 08) profit (p. 3) quadratic approimation (p. 45) related rates (p. 46) relative change (p. 40) relative etrema (p. 89) Rolle s Theorem (p. 96) second derivative test for local etrema (p. ) standard linear approimation (p. 33)

71 56 Chapter 4 Applications of Derivatives Chapter 4 Review Eercises The collection of eercises marked in red could be used as a chapter test. In Eercises and, use analtic methods to find the global etreme values of the function on the interval and state where the occur..,. 3 9, In Eercises 3 and 4, use analtic methods. Find the intervals on which the function is (a) increasing, (b) decreasing, (c) concave up, (d) concave down. Then find an (e) local etreme values, (f) inflection points. 3. e 4. 4 In Eercises 5 6, find the intervals on which the function is (a) increasing, (b) decreasing, (c) concave up, (d) concave down. Then find an (e) local etreme values, (f) inflection points e cos ln,, 0. sin 3 cos 4, 0 p e 3., 0 { 4 3, In Eercises 3 and 4, find the function with the given derivative whose graph passes through the point P. 3. f sin cos, P p, 3 4. f 3, P, 0 In Eercises 5 and 6, the velocit v or acceleration a of a particle is given. Find the particle s position s at time t. 5. v 9.8t 5, s 0 when t 0 6. a 3, v 0 and s 5 when t 0 In Eercises 7 30, find the linearization L of f at a. 7. f tan, a p 4 8. f sec, a p 4 9. f, a f e tan sin, a 0 In Eercises 3 34, use the graph to answer the questions. 3. Identif an global etreme values of f and the values of at which the occur. O (, ) f (), Figure for Eercise 3 Figure for Eercise 3 3. At which of the five points on the graph of f shown here (a) are and both negative? (b) is negative and positive? 33. Estimate the intervals on which the function f is (a) increasing; (b) decreasing. (c) Estimate an local etreme values of the function and where the occur. O f() P Q (, 3) R S T In Eercises 7 and 8, use the derivative of the function f to find the points at which f has a (a) local maimum, (b) local minimum, or (c) point of inflection ( 3, ) f'() In Eercises 9, find all possible functions with the given derivative. 9. f 5 e 0. f sec tan. f, 0. f 34. Here is the graph of the fruit fl population from Section.4, Eample. On approimatel what da did the population s growth rate change from increasing to decreasing?

72 Review Eercises 57 Number of flies p Time (das) 35. Connecting f and f The graph of f is shown in Eercise 33. Sketch a possible graph of f given that it is continuous with domain 3, and f Connecting f, f, and f The function f is continuous on 0, 3 and satisfies the following. 0 3 f f 3 0 does not eist 4 f 0 does not eist f f f (a) Find the absolute etrema of f and where the occur. (b) Find an points of inflection. (c) Sketch a possible graph of f. 37. Mean Value Theorem Let f ln. (a) Writing to Learn Show that f satisfies the hpotheses of the Mean Value Theorem on the interval a, b 0.5, 3. (b) Find the value(s) of c in a, b for which f c f b f a. b a (c) Write an equation for the secant line AB where A a, f a and B b, f b. (d) Write an equation for the tangent line that is parallel to the secant line AB. 38. Motion along a Line A particle is moving along a line with position function s t 3 4t 3t t 3. Find the (a) velocit and (b) acceleration, and (c) describe the motion of the particle for t Approimating Functions Let f be a function with f sin and f 0. (a) Find the linearization of f at 0. (b) Approimate the value of f at 0.. (c) Writing to Learn Is the actual value of f at 0. greater than or less than the approimation in (b)? t 40. Differentials Let e. Find (a) d and (b) evaluate d for and d Table 4.5 shows the growth of the population of Tennessee from the 850 census to the 90 census. The table gives the population growth beond the baseline number from the 840 census, which was 89,0. Table 4.5 Population Growth of Tennessee Years since Growth Beond Population 0 73, , , , ,308 60,9,406 70,355,579 Source: Bureau of the Census, U.S. Chamber of Commerce (a) Find the logistic regression for the data. (b) Graph the data in a scatter plot and superimpose the regression curve. (c) Use the regression equation to predict the Tennessee population in the 90 census. Be sure to add the baseline 840 number. (The actual 90 census value was,337,885.) (d) In what ear during the period was the Tennessee population growing the fastest? What significant behavior does the graph of the regression equation ehibit at that point? (e) What does the regression equation indicate about the population of Tennessee in the long run? (f) Writing to Learn In fact, the population of Tennessee had alread passed the long-run value predicted b this regression curve b 930. B 000 it had surpassed the prediction b more than 3 million people! What historical circumstances could have made the earl regression so unreliable? 4. Newton s Method Use Newton s method to estimate all real solutions to cos 0. State our answers accurate to 6 decimal places. 43. Rocket Launch A rocket lifts off the surface of Earth with a constant acceleration of 0 m sec. How fast will the rocket be going min later? 44. Launching on Mars The acceleration of gravit near the surface of Mars is 3.7 m sec. If a rock is blasted straight up from the surface with an initial velocit of 93 m sec (about 08 mph), how high does it go? 45. Area of Sector If the perimeter of the circular sector shown here is fied at 00 ft, what values of r and s will give the sector the greatest area? r r s

73 58 Chapter 4 Applications of Derivatives 46. Area of Triangle An isosceles triangle has its verte at the origin and its base parallel to the -ais with the vertices above the ais on the curve 7. Find the largest area the triangle can have. 47. Storage Bin Find the dimensions of the largest open-top storage bin with a square base and vertical sides that can be made from 08 ft of sheet steel. (Neglect the thickness of the steel and assume that there is no waste.) 48. Designing a Vat You are to design an open-top rectangular stainless-steel vat. It is to have a square base and a volume of 3 ft 3 ; to be welded from quarter-inch plate, and weigh no more than necessar. What dimensions do ou recommend? 49. Inscribing a Clinder Find the height and radius of the largest right circular clinder that can be put into a sphere of radius 3 as described in the figure. 5. Inscribing a Rectangle A rectangle is inscribed under one arch of 8 cos 0.3 with its base on the -ais and its upper two vertices on the curve smmetric about the -ais. What is the largest area the rectangle can have? 53. Oil Refiner A drilling rig mi offshore is to be connected b a pipe to a refiner onshore, 0 mi down the coast from the rig as shown in the figure. If underwater pipe costs $40,000 per mile and land-based pipe costs $30,000 per mile, what values of and give the least epensive connection? mi Rig 0 Refiner 0 mi h r Designing an Athletic Field An athletic field is to be built in the shape of a rectangle units long capped b semicircular regions of radius r at the two ends. The field is to be bounded b a 400-m running track. What values of and r will give the rectangle the largest possible area? 50. Cone in a Cone The figure shows two right circular cones, one upside down inside the other. The two bases are parallel, and the verte of the smaller cone lies at the center of the larger cone s base. What values of r and h will give the smaller cone the largest possible volume? 55. Manufacturing Tires Your compan can manufacture hundred grade A tires and hundred grade B tires a da, where 0 4 and Your profit on a grade A tire is twice our profit on a grade B tire. What is the most profitable number of each kind to make? ' r 56. Particle Motion The positions of two particles on the s-ais are s cos t and s cos t p 4. (a) What is the farthest apart the particles ever get? (b) When do the particles collide? 6' h 57. Open-top Bo An open-top rectangular bo is constructed from a 0- b 6-in. piece of cardboard b cutting squares of equal side length from the corners and folding up the sides. Find analticall the dimensions of the bo of largest volume and the maimum volume. Support our answers graphicall. 5. Bo with Lid Repeat Eercise 8 of Section 4.4 but this time remove the two equal squares from the corners of a 5-in. side. 58. Changing Area The radius of a circle is changing at the rate of p m sec. At what rate is the circle s area changing when r 0 m?