i 3. Find a formula for f'(x).

Transcription

1 ln Section 4.1 Etreme Values of Functions 183 Eample 6 USING GRAPHICAL METHODS Find the etreme values of f() = In 11 : 21. Maimum X = [-4.5, 4.5] b [-4, 2] Y = '-172 Figure 4.9 The function in Eample 6. Solution Solve Graphicall The domain of f is the set of all nonzero real numbers. Figure 4.9 suggests thatfis an even function with a maimum value at two points. The coordinates found in this window suggest an etreme value of about at approimatel = 1. Because f is even, there is another etreme of the same value at approimatel = -1. The figure also suggests a minimum value at = 0, but f is not defined there. Confirm The derivative Analticall 1-2 f'() = (1 + 2) is defined at ever point of the function's domain. The critical points where f' () = 0 are = 1 and = - 1. The corresponding values of I are both In (1/2) = + 2 = Finding Etreme Values Let I() = I2 : 11, -2 -s :s Determine graphicall the etreme values of I and where the I occur. Findf' at these values of. 2. Graphfandf' (or NDER (f(),, )) in the same viewing n! window, Comment on the relationship between the graphs. i 3. Find a formula for f'(). ;L- Quick Review 4a1 In Eercises 1-8, find the first derivative of the function. I.f()=~ 2 3. f() =, r,::--o v g() = In ( 2 + 1) 7. he) = e 2 :c 2. f() = 3 / f() =.3r:;--:- \I g() = cas (In ) 8. he) = e 1n In Eercises 11 and 12, let f() = {3-2, :S 2 + 2, > Find (a) 1'0), (b) 1'(3), (c) f'(2). 12. (a) Find the domain of 1'. (b) Write a formula for f'(). In Eercises 9 and 10, find the limit for f() 2 =, r,::--o' v9 - X1 9. lim f() 10. lim f() -;3 - -;- 3+

2 184 Chapter 4 Applications of Derivatives. Section 491 Eercises In Eercises 1-6, identif each -value at which an absolute etreme value occurs. Eplain how our answer is consistent with the Etreme Value Theorem ~O~~a--~c----L-+X 5. <: ~a--~c-----b~ \:;' ~--~~----~_ o a c b o a c -04-~a--~b--~c---+ = g() In Eercises 7-10, find the <.treme values and where the occur Y f() = sin ( + :), o s ::s 7; 11' 311' 16. g()= see, -- < < f() = 2/5, -3 ::s < I 18. f() = 3/5, -2 < ::s 3 In Eercises 19-30, find the etreme values of the function and where the occur. 19. = = =~ Y = r:----;; v I Y = Y Y = Y = Y = Y = = Y =.3r:----;; 'V Y = In Eercises 31-34, work in groups of two or three to find the etreme values of the function on the interval and where the occur. 31. f() = I [ + 31, -5::S X::S g() = I [ - 51, -2::s ::s7 ----~-+--~--~ --~--~----~-+ -2 o he) = [ [ - 31, -00 < < k() = I I - 31, -00 < < ' ''---+ o 2 X In Eercises 11-18, use analtic methods to find the etreme values of the function on the interval and where the occur f() = - + In, 0.5::s ::s g() = e-, -1::s ::s he) = In ( + 1), O::s ::s k() = e=', -00 < < 00-1 Eplorations In Eercises 35 and 36,give reasons for our answers. 35. Writing to learn Let f() = ( - 2)2/3. (a) Does 1'(2) eist? (b) Show that the onl local etreme value of f occurs at = 2. (e) Does the result in (b) contradict the Etreme Value Theorem? (d) Repeat (a) and (b) for f() = ( - a)2/3, replacing 2 b a. 36. Writing to learn Let f() = I 3-9l. (a) Does 1'(0) eist? (b) Does 1'(3) eist? (e) Does 1'(-3) eist? (d) Determine all etrema of f II "

3 Section 4.1 Etreme Values of Functions 185 In Eercises 37-44, find the derivative at each critical point and determine the local etreme values. 37. =' X 2 / 3 (X + 2) 38. =' 2 / 3 ( 2-4) 39.Y=~ 4-2X ~l 41. Y =' ( + 1. ' > Y =' (~ ~ ;~ _ 2, ~ ~ ~ , $ = ( _ , > 1 l-l.2 _l =' ' , In Eercises 45-48, 45. f'() Y='X2~ ~l >l match the table with a graph off(). 46. f'() a 0 a 0 b 0 b 0 e 5 e -5 f'() 48. f'() a does not eist a does not eist b 0 b does not eist e -2(' e -1.7 a b c 50. Writing to Learn The function 200 P() = 2 + -, 0 < < 00, models the perimeter of a rectangle of dimensions b 100/. (a) Find an etreme values of P. (b) Give an interpretation in terms of perimeter of the rectangle for an values found in (a). Etending the Ideas 51. Cubic Functions Consider the cubic function f() = a 3 + b 2 + e + d. (a) Show thatf can have 0, 1, or 2 critical points. Give eamples and graphs to support our argument. (b) How man local etreme values can f have? 52. Proving Theorem 2 Assume that the function f has a local maimum value at the interior point e of its domain. (a) Show that there is an open interval containing f() - fee) $ 0 for all in the open interval. (b) Writing to Learn (e) Writing to Learn (d) Writing to Learn conclude f'(e) = O. Now eplain wh we ma sa lim f() - fee) '$ o. X--7C+ X - C Now eplain wh we ma sa lim_ f() - f(~: 2: O. -tc X -. eft; e such that Eplain how (b) and (c) allow us to (e) Writing to Learn Give a similar argument if f has a local minimum value at an interior point. (a) (b) In Eercise 53, work in groups of two or three. 53. Functions with No Etreme Values at Endpoints (a) Graph the function a b c (c) (d) 49. Writing to Learn The function Ve) = (10-2)(l6-2), 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. f() = Isin~, > 0 0, = O. Eplain wh f(o) = 0 is not a local etreme value of f (b) Construct a function of our own that fails to have an etreme value at a domain endpoint..

4 192 Chapter 4 Applications of Derivatives Section 4.2 Eercises In Eercises 1-8, 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. 1. f() = he) =- 5. f() = e 2 7. = 4 - V+2 2. g() = Z k() = 2" 6. f() = e- O = X4 - lo In Eercises 9-14, find (a) the local etrema, (b) the intervals on which the function is increasing, and (e) the intervals on which the function is decreasing. 9.f() =~ 11. h()=~ f() = cos 10. g() = 1 / 3 ( + 8) 12. k() = g() = 2 + cos In Eercises 15-18, (a) show that the function f satisfies the hpotheses of the Mean Value Theorem on the given interval [a, b]. (b) Find each value of e in (a, b) that satisfies the equation f'(c) = f(b~ = ~(a). 15. f() = , [0, IJ 16. f() = 2/3, [0, IJ 17. f() = sin- 1, [-1,1] 18. f() = In ( - 1), [2,4] In Eercises 19 and 20, the interval a:s :S 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. 19. f() 1 = + -, 0.5:S cs f() =~, 1:S X :S 3 In Eercises 21-24, (a) show that the function f does not satisf the hpotheses of the Mean Value Theorem on the given interval [a, b]. (b) Graphftogether with the line through the points A(a, f(a)) and Bib, f(b»). (e) Find an values of e in (a, b) that satisf the equation j'(e) = f(b~ = ~(a). 21. f() = X ll3, [-I,IJ 22. f() = [ - 11, [0,3J 23. f() = 1 -li, [-1,1] 24. f() = {cas,. -7T :S < 0, on [-7T, 7T] 1 + sin, O:S :S 7T, In Eercises 25-30, find all possible functions f with the given derivative. 25. f'e) = 27. f'e) = f'e) = ex 26. j'() = f'e) = sin 30. j'() = _1-1' > 1 - In Eercises 31-34, find the function with the given derivative whose graph passes through the point P. 31. f'e) = -~, > 0, P(2,1) 32. f'e) = 4: 3/4 ' P(1, -2) 33. f'e) =! 2' > -2, P(-1,3) 34. j'() = cas, P(0,3) In Eercises 35-38, work in groups of two or three and sketch a graph of a differentiable function =f() that has the given properties. 35. (a) local minimum at (1, 1), local maimum at (3, 3) (b) local minima at (1, 1) and (3, 3) (c) local maima at (1, 1) and (3, 3) 36. f(2) = 3, 1'(2) = 0, and (a) j'() > for < 2, j'() < for > 2. (b) j'() < for < 2, j'() > for > 2. (e) f'e) < 0 for *- 2. (d) f'e) > 0 for * '(-1) =1'(1) = 0, f'() > on (-1, 1), f'() < for < -1, f'e) > for > A local minimum value that is greater than one of its local maimum values 39. Speeding A trucker handed in a ticket at a toll booth showing that in 2 h she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Wh? 40. Temperature Change It took 20 see for: the temperature to rise from O F to 212 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 F/sec. 41. Triremes Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 h. Eplain wh at some point during this feat the trireme's speed eceeded 7.5 knots (sea miles per hour). 42. Running a Marathon A marathoner ran the 26.2-mi New York Cit Marathon in 2.2 h. Show that at least twice, the marathoner was running at eactl 11 mph.

5 Section 4.2 Mean Value Theorem Free Fall On the moon, the acceleration due to gravit is 1.6 m/sec" (a) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 see 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? r 44. Diving (a) With what velocit will ou hit the water if ou step off from a 10-m diving platform? (b) With what velocit will ou hit the water if ou dive off the platform with an upward velocit of 2 m/sec? Eplorations 51. Analzing Derivative Data Assume thatfis continuous on [-2,2J and differentiable on (-2, 2). The table gives some values of f'(). f'() f'() l (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. (e) Use the model in (b) for f' and find a formula for f that satisfies f(o) == O. 52. Analzing Motion Data Pria's distance D in meters from a motion detector is given b the data in Table 4.1. Table 4.1 Motion Detector Data 45. Writing to Learn The function f() = {. O:so; < 1 0, = 1 is zero at = 0 and at = 1. Its derivative is equal to 1 at ever point between 0 and I, so f' is never zero between 0 and 1, and the graph of f has no tangent parallel to the chord from (0, 0) to (1, 0). Eplain wh this does not contradict the Mean Value Theorem. 46. Writing to Learn Eplain wh there is a zero of == cas between ever two zeros of == sin. 47. Unique Solution Assume thatfis continuous on [a, bj and differentiable on (a, b). Also assume thatf(a) andf(b) have opposite signs and f' =1= 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.) 48. X = 0, -2:so; :so; In ( + 1) = 0, O:so; :so;3 50. Parallel Tangents Assume that f and g are differentiable on [a, bj 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. t (sec) D (m) t (see) 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. (e) Find a cubic regression equation D = f(t) for the data in Table 4.1 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).

6 :f~ r 1I Section 4.3 Connecting I' and 1" with the Graph off 203 f j I Ị. t t I I r I! I I Quick Review 4.3 In Eercises 1 and 2, factor the epression and use sign charts to solve the inequalit < > 0 In Eer?ises 3-6, find the domains of f and I'. 3. f() = e" 4. f() = X 3 / 5 S. f() = ~2 6. f() = 2 / 5 - In Eercises 7-10, find the horizontal asmptotes of the function's graph. 7. = (4-2)e = 1 + toe-o.5 8. = ( 2 - )e = 2 + 5e-O.1 Section 4.3 Eercises In Eercises 1 and 2, use the graph of the function f to estimate where (a) r and (b) f" are 0, positive, and negative. 1. Y 2. Y Y =f() =f() In Eercises 3-6, use the graph off' to estimate the intervals on which the function f is (a) increasing or (b) decreasing. (c) Estimate where f has local etreme values: 3. Y 4. Y Y =f'() 2

7 204 Chapter 4 Applications of Derivatives 5. The domain of t' is [0,4) U (4,6]. 21. = ei!2 23. = tan " 25. = 1 / 3 (X - 4) I 27. = Y=X2~ 24. = X 3 / 4 (5 - ) 26. = 1 / 4 ( + 3) 28. = =f'() / 6. The domain of f' is [0, 1) U (1,2) U (2,3]. In Eercises 29 and 30, 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. 29. ' = ( - 1)2( - 2) 30. ' = ( - 1)2(X - 2)( - 4) Eercises 31 and 32 show the graphs of the first and second derivatives of a function =f(). Work in groups of two or three. Cop the figure and add a sketch of a possible graph of f that passes through the point P. 31. ~ -L J- L- ~ \-----c~--+_--~ In Eercises 7-12, use analtic methods to find the intervals on which the function is (a) increasing, (c) concave up, Then find an (e) local etreme values, Support our answers graphicall. 7. = 2 - X = =~ (b) decreasing, (d) concave down. (f) inflection points. 8. = = e 1 / 12. = {32- l 2, X < 0 +, :2:: 0 In Eercises 13-28, find the intervals on which the function is (a) increasing, (c) concave up, Then find an (e) local etreme values, 13. = l Y = 2X I / = 5e X ex + 3eO.8 <l :2::1 (b) decreasing, (d) concave down. (f) inflection points. 14. = = 5 - I!3 18 _ 8e- X - 2e-X + 5e p =f'() -*-----~----~----~----~-+ In Eercises 33 and 34, work in groups of two or three. (a) Find the absolute etrema of f and where the occur. (b) Find an points of inflection. (c) Sketch a possible graph off 33. f is continuous on [0, 3J and satisfies the following f r 3 0 does not eist -3 fn 0-1 does not eist 0 0<<1 1<<2 2<<3 f r f" - - -

8 Section 4.3 Connecting I' and f" with the Graph off fis an even function, continuous on [-3, 3], and satisfies the following f I' does not eist 0 does not eist f" does not eist 0 does not eist O<<1 l<<2 2<<3 f I' f" (d) What can ou conclude about f(3) and f(-3)? In Eercises 35 and 36, work in groups of two or three. Sketch a possible graph of a continuous function f that has the given properties. 35. Domain [0, 6], graph of I' given in Eercise 5, and f(o) = Domain [0, 3], graph of I' given in Eercise 6, and f(o) = -3. I _ In Eercises 37-40, a particle is moving along a line with position function set). Find the (a) velocit and (b) acceleration, and (c) describe the motion of the particle for t 2: O. 37. set) = t 2-4t set) = 6-2t - t set) = t 3-3t set) = 3t 2-2t 3 In Eercises 41 and 42, the graph of the position function = set) 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? 41. d).5 ubi) c:0c:; ee 0.11i E Ao -= 42. Time (see) = s(t) 43. Writing to Learn If f() is a differentiable function and f'(e) = 0 at an interior point e off's domain, must f have a local maimum or minimum at = e? Eplain. 44. Writing to Learn If f() is a twice-differentiable function and I"(e) = 0 at an interior point e of f's domain, mustf have an inflection point at = e? Eplain. 45. Connecting f and f' Sketch a smooth curve = f() through the origin with the properties that f'e) < 0 for < 0 and f'e) > 0 for > O. 46. Connecting f and f" Sketch a smooth curve = fe) through the origin with the properties that re) < 0 for < 0 and re) > 0 for > O. 47. Connecting f, f', and f" Sketch a continuous curve =f() with the following properties. Label coordinates where possible. f(-2) = 8 f(o) = 4 f(2) = 0 1'(2) = 1'(-2) = 0 f'() > 0 for [] > 2 f'() < 0 for [] < 2 re) < 0 for < 0 re) > 0 for > Using Behavior to Sketch Sketch a continuous curve = f() with the following properties. Label coordinates where possible. Curve <2 falling, concave up 2 1 horizontal tangent 2<<4 4<<6 rising, concave up 4 4 inflection point rising, concave down 6 7 horizontal tangent >6 falling, concave down 49. Table 4.3 gives the sales of prepaid phone cards for several ears (the values for 1997 and 1998 were estimated). Let = 0 represent 1992, = 1 represent 1993, and so forth. (a) Find the logistic regression equation and superimpose its graph on a scatter plot of the data. (b) Use the regression equation to predict when the rate of increase in sales will start to decrease, and to predict the sales at that time. (c) At what amount does the regression equation predict that sales will stabilize? Table 4.3 Year Prepaid Phone Card Sales Sales (millions of dollars) Source: Atlantic ACM as reported b Grant Jerding in USA Toda, April 16, 1997.

9 214 Chapter 4 Applications of Derivatives Quick Review 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 S em and height 8 ern. 4. Find the dimensions of a right circular clinder with volume 1000 cm ' and surface area 600 em", In Eercises S-8, rewrite the epression as a trigonometric function of the angle a. 5. sin (-a) 7. sin (7T - a) 6. cos (-a) 8. cos (7T - a) In Eercises 9 and 10, use substitution to find the eact solutions of the sstem of equations. I2 2 X2 + 2 = 4 -+-= { =V3 =+3 Section 4m4 Eercises In Eercises 1-10, solve the problem analticall. Support our answer graphicall. 1. Finding Numbers The sum of two nonnegative numbers is 20. 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. 2. Maimizing Area What is the largest possible area for a right triangle whose hpotenuse is S em long, and what are its dimensions? 3. Maimizing Perimeter What is the smallest perimeter possible for a rectangle whose area is 16 in2, 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 2 units long. (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? is 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 g. b IS-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 comer of the first quadrant with a line segment 20 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? 10. The Shortest Fence A 216-m2 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? 11. Designing a Tank Your iron works has contracted to design and build a SOO-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 us~? (b) 'Writing to Learn Briefl describe how ou took weight into account. -1

10 Section 4.4 Modeling and Optimization Catching Rainwater A 1125-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 ( 2 +.4) + lo, what values of and will minimize (b) Writing to Learn function in (a). 18. Designing a Bo with Lid A piece of cardboard measures 10- b IS-in. Two equal squares are removed from the corners of a lo-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the it? Give a possible scenario for the cost 13. 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 2-in. margin at tach side. What overall dimensions will minimize the amount of paper used? 14. Vertical Motion The height of an object moving verticall is given b s = -16t t + 112, 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 (e) its velocit when s = O. 15. Finding an Angle Two sides of a triangle have lengths a and b, and the angle between them is e. What value of e will maimize the triangle's area? (Hint: A = 0/2) ab sin e.) 16. Designing a Can What are the dimensions of the lightest open-top right circular clindrical can that will hold a volume of 1000 cm '? Compare the result here with the result in Eample Designing a Can You are designing a 1000-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 2r units on a side. The total amount of aluminum used up b the can will therefore be A = 8r Trh rather than the A = 27Tr2 + 27Trh in Eample 2. In Eample 2 the ratio of h to r for the most economical can was 2 to 1. What is the ratio now? In Eercises 18 and 19, work in groups of two or three. tabs can be folded to form a rectangular bo with lid. '\ Quickest Route Jane is 2 ~i offshore in a boat and wishes ( ) W 't f I V()"' th I f th b Ii.:.~\j/l / a n e a ormu a ror e vo ume a e o. to reach a coastal VIllage 6 rru down a straight shoreline '!i' (, J from the point nearest the boat. She can row 2 mph and can (b) Find the domain of V for the problem situation and 'l'- n 1; walk 5 mph. Where should she land her boat to reach the graph V over this domain. J\" village in the least amount of time? (e) Use a graphical method to find the maimum volume..andjhevalue of that gives it. (d) Confirm our result in (c) analticall. T.-:~_-,- -,f 19. Designing a Suitcase A 24- b 36-in. sheet of cardboard is folded in half to form a 24- b I8-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. (a) Write a formula T I Ve) for the volume of the bo. (b) Find the domain of V for the problem situation and graph V over this domain. (e) Use a graphical method to find the maimum volume and the value of that gives it. (d) Confirm our result in (c) analticall. (e) Find a value of that ields a volume of 1120 in 3 (f) Writing to Learn Write a paragraph describing the issues that arise in part (b). 124" 1k-~( "----~)1 ~18"--4 The sheet is then unfolded. 124" Base 1f<-1(----36"----.;>J)1 124" Inscribing Rectangles A rectangle is to be inscribed under the arch.of.jhe.curze ~",Acos.(O.S) from = -7T to X = 7T. Wbat are the dimensions of the rectangle with largest area, and what is the largest area?

11 216 Chapter 4 Applications of Derivatives 22. Maimizing Volume Find the dimensions of a right circular clinder of maimum volume that can be inscribed in a sphere of radius 10 ern. What is the maimum volume? Eploration 23. The figure shows the graph of f() = e :", 2: O. 26. Constructing Clinders Compare the answers to the following two construction problems. (a) A rectangular sheet of perimeter 36 em and dimensions em b em 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? ~>< a b (a) Find where the absolute maimum of f occurs. (b) Let a> 0 and b> 0 be given a~ shown in the figure. Complete the following table where A is the area of the rectangle in the figure. Circumference = a b A (a) 27. Constructing Cones A right triangle whose hpotenuse is V3 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. (b) (c) Draw a scatter plot of the data (a, A). (d) Find the quadratic, cubic, and quartic regression equations for the data in (b), and superimpose their graphs on a scatter plot of the data. (e) Use each of the regression equations in (d) to estimate the maimum possible value of the area of the rectangle. Iii! 24. Cubic Polnomial Functions Let f() = a 3 + b 2 + c + d, a *" O. (a) Show that f has either 0 or 2 local etrema. (b) Give an eample of each possibilit in (a). 25. 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 108 in. What dimensions will give a bo with a square end the largest possible volume? 28. Finding Parameter Values What value of a makes f() = X2 + (al) have (a) a local minimum at = 2? (b) a point of inflection at = I? 29. Finding Parameter Values Show that f() = X2 + (al) cannot have a local maimum for an value of a. 30. Finding Parameter Values What values of a and b make f() = 3 + a 2 + b have (a) a local maimum at = -1 and a local minimum at = 3? (bja local minimum at = 4 and a point of inflection at = I? 31. Inscribing a Cone Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.

12 Section 4.4 Modeling and Optimization 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 12-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 = 1. Reconcile what ou see with our answer in (a). (c) Writing to learn On 'the same screen, graph S as a function of the beam's depth d, again taking k = 1. Compare the graphs with one another and with our answer in (a). What would be the effect of changing to some other value of k? Tr it. 35. Electrical Current Suppose that at an time t (see) the current i (amp) in an alternating current circuit is i = 2 cos t + 2 sin t. What is the peak (largest magnitude) current for this circuit? 36. Calculus and Geometr How close does the curve = V come to the point (3/2, OJ? (Hint: If ou minimize the square of the distance, ou can avoid square roots.) --~~------~ ~X o 0,0) 37. Calculus and Geometr How close does the semicircle = ~ come to the point (1, V3)? 33. 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 12-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 = 1. Reconcile what ou see with our answer in (a). (c) Writing to Learn On the same screen, graph S as a function of the beam's depth d, again taking k = 1. Compare the graphs with one another and with our answer in (a). What would be the effect of changing to some other value of k? Tr it. 34. Frictionless Cart A small frictionless cart, attached to the wall b a spring, is pulled 10 em 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 = 10 cos -m. (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? 38. Writing to learn Is the function j() = X2 - X + 1 ever negative? Eplain. 39. Writing to learn You have been asked to determine whether the function j() = cos + cos 2 is ever negative. (a) Eplain wh ou need to consider values of onl in the interval [0, 271']. (b) Is j ever negative? Eplain. 40. Vertical Motion Two masses hanging side b side from springs have positions SI = 2 sin t and S2 = sin 2t, respectivel, with SI and S2 in meters and t in seconds. (a) At what times in the interval t> do the masses pass each other? (Hint: sin 2t = 2 sin t cos t.) (b) When in the interval O:S i s: 271' is the vertical distance between the masses the greatest? What is this distance? (Hint: cos 2t = 2 cos? t - 1.),i.. ~, -:>,; '~~" ~ s

13 r~~ i 218 Chapter 4 Applications of Derivatives 41. Motion on a Line The positions of two particles on the s-ais are SI = sin t and Sz = sin (r + 71"/3), with SI and S2 in meters and t in seconds. (a) At what time(s) in the interval particles meet? 0:::; i s: 271"do the (b) What is the farthest apart that the particles ever get? (c) When in the interval 0:::; i s: 271"is the distance between the particles changing the fastest? 42. Finding an Angle The trough in the figure is to be made to the dimensions shown: Onl the angle 8 can be varied. What value of e will maimize the trough's volume? 46. 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 1 Normal Angle of reflection 2 Light receiver B Plane mirror ~! 43. Paper Foldjng Work in groups of two or three. A rectangular sheet of 8 1/2- b l l-in. paper is placed on a flat surface. One of the comers 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 L2 = 2 3 /(2-8.S). (b) What value of minimizes L2? (c) What is the minimum value of L? 44. Sensitivit to Medicine (continuation of Eercise 35, 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. 45. 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(loo - ), -c where a and b are certain positive constants. What selling price will bring a maimum profit? 47. Tin Pest When metallic tin is kept below 13.2 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. In some cases it is reasonable to assume that the rate v = dldt of the reaction is proportional both to the amount ofthe original substance present and to the amount of product. That is, v ma be considered to be a function of alone, and v = kta - ) = ka r: k2, where = the amount of product, a = the amount of substance k = a positive constant. at the beginning, At what value of does the rate v have a maimum? What is the maimum value of v?

14 Section 4.4 Modeling and Optimization 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 ern/see) can be modeled b the equation where ro is the rest radius of the trachea in em and e is a positive constant whose value depends in part on the length of the trachea. (a) Show that v is greatest when r = (2/3)ro, that is, when the trachea is about 33% contracted. The remarkable fact is that Xsra photographs confirm that the trachea contracts about this much during a cough. (b) Take ro to be 0.5 and e to be 1, and graph v over the interval 0:5 r: Compare what ou see to the claim that v is a maimum when r = (2/3)ro. 49. Tour Service You operate a tour service that offers the following rates: $200 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 $2. It costs $6000 (a fied cost) plus $32 per person to conduct the tour. How man people does it take to maimize our profit? 50. 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) km hq =q+em + T' 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), e 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? 51. Production Level Show that if re) = 6 and c() = are our revenue and cost functions, then the best ou can do is break even (have revenue equal cost). 52. Production Level Suppose c() = ,000 is the cost of manufacturing items. Find a production level that will minimize the average cost of making items. Etending the Ideas 53. 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 2 + e + d where (-L) = Hand (O) = O. (a) What is d/d at = O? (b) What is d/d at = -L? (e) Use the values for d/d at = 0 and = -L together with (O) = 0 and (-L) = H to show that Landing path H = Cruising altitude Airport ---i ~~=--~~~ ~I( L ~ In Eercises 54 and 55, ou might find it helpful to use a CAS and to work in groups of two or three. 54. Generalized Cone Problem A cone of height hand radius r is constructed from a flat, circular disk of radius a in. as described in Eploration 1. (a) Find a formula for the volume V of the cone in terms of and a. (b) Find rand h in the cone of maimum volume for a = 4, 5,6,8. (e) Writing to Learn Find a simple relationship between rand h that is independent of a for the cone of maimum volume. Eplain how ou arrived at our relationship.