HW 2 Date: Name Find the relative extrema of the function, if they exist. 1) f(x) = x2-4x + 7 1)

Transcription

1 HW 2 Date: Name Find the relative etrema of the function, if the eist. 1) f() = ) 2) s() = ) 3) f() = ) 4) f() = ) 5) f() = ) 6) f() = ) 7) f() = ) 8) f() = ) 4 9) f() = 2-1 9) 8 10) f() = ) 11) f() = ) 12) f() = ) Find the relative etrema of the function and classif each as a maimum or minimum. 13) f() = ) 14) f() = ) 15) s() = ) 16) = ) 17) f() = ) 18) f() = ( + 3) 3 18) 1r

2 19) f() = ) Solve the problem. 20) A firm estimates that it will sell N units of a product after spending dollars on advertising where N() = , 0 500, and is in thousands of dollars. Find the relative etrema of the function. 21) Assume that the temperature of a person during an illness is given b T(t) = -0.1t t , 0 t 14, where T = the temperature ( F) at time t, in das. Find the relative etrema of the function. 22) The Olmpic flame at the 1992 Summer Olmpics was lit b a flaming arrow. As the arrow moved d feet horizontall from the archer, assume that its height h, in feet, was approimat the function h = d d Find the relative maimum of the function. 20) 21) 22) Graph the function b first finding the relative etrema. 23) f() = ) 24) f() = ) 25) f() = ) 26) f() = ) Draw a graph to match the description. Answers will var. 27) f() is decreasing over (-, 4] and increasing over [4, ). 27) 28) f() has a negative derivative over (-, -9) and a positive derivative over (-9, ). 28) 29) G() has a positive derivative over (-, -7) and (-3, 7) and a negative derivative over (-7, - 3) and (7, ). 29) 2

3 Graph the function. 30) f() = ) 31) f() = ) 32) f() = ) Find the points of inflection. 33) f() = ) 34) f() = ) 35) f() = ) 36) f() = ) 37) f() = ) Determine where the given function is increasing and where it is decreasing. 38) s() = ) 39) f() = ) 40) = ) 41) f() = ) Determine where the given function is concave up and where it is concave down. 42) f() = ) 43) q() = ) 44) f() = ) 45) f() = ) 3

4 Draw a graph to match the description. Answers will var. 46) f() is decreasing and concave up on ( -, -10); f() is decreasing and concave down on (- 10, ). 46) 47) f() is increasing and concave up on (-, -5); f() is increasing and concave down on (-5, ). 47) 48) f() is decreasing and concave down on (-, 10); f() is decreasing and concave up on (10, ). 48) 49) f() is increasing and concave down on (-, 8); f() is increasing and concave up on (8, ). 49) Solve the problem. 50) The annual revenue and cost functions for a manufacturer of grandfather clocks are approimatel R() = and C() = ,000, where denotes the number of clocks made. What is the maimum annual profit? 51) The annual revenue and cost functions for a manufacturer of precision gauges are approimatel R() = and C() = ,000, where denotes the number of gauges made. What is the maimum annual profit? 52) Because of material shortages, it is increasingl epensive to produce 6.0L diesel engines. In fact, the profit in millions of dollars from producing hundred thousand engines is approimated b P() = , where Find the inflection point of this function to determine the point of diminishing returns. 50) 51) 52) 53) The function R() = , 0 20, represents revenue in thousands of dollars where represents the amount spent on advertising in tens of thousands of dollars. Find the inflection point for the function to determine the point of diminishing returns. 53) Find the absolute maimum and absolute minimum values of the function, if the eist, over the indicated interval. When no interval is specified, use the real line ( -, ). 54) f() = -21; [-7, 7] 54) 55) f() = 6 + 2; [-1, 2] 55) 56) f() = ; [0, 5] 56) 4

5 57) f() = ; [-8, 8] 57) Find the absolute maimum and absolute minimum values of the function, if the eist, on the indicated interval. 58) f() = ; [-9, 0] 58) Find the absolute maimum and absolute minimum values of the function, if the eist, over the indicated interval. When no interval is specified, use the real line ( -, ). 59) f() = ; (0, ) 59) Solve the problem. 60) A carpenter is building a rectangular room with a fied perimeter of 420 ft. What are the dimensions of the largest room that can be built? What is its area? 60) 61) Find the dimensions that produce the maimum floor area for a one -stor house that is rectangular in shape and has a perimeter of 166 ft. Round to the nearest hundredth, if necessar. 61) 62) An architect needs to design a rectangular room with an area of 83 ft2. What dimensions should he use in order to minimize the perimeter? Round to the nearest hundredth, if necessar. 62) 63) A farmer decides to make three identical pens with 136 feet of fence. The pens will be net t each other sharing a fence and will be up against a barn. The barn side needs no fence. 63) What dimensions for the total enclosure (rectangle including all pens) will make the area as as possible? 64) A compan wishes to manufacture a bo with a volume of 40 cubic feet that is open on top and is twice as long as it is wide. Find the width of the bo that can be produced using the minimum amount of material. Round to the nearest tenth, if necessar. 64) 65) Find the number of units that must be produced and sold in order to ield the maimum pr given the following equations for revenue and cost: R() = C() = ) 5

6 66) Find the number of units that must be produced and sold in order to ield the maimum pr given the following equations for revenue and cost: R() = 7 C() = ) 67) An appliance compan determines that in order to sell dishwashers, the price per dishwas must be p = It also determines that the total cost of producing dishwashers is given b C() = How man dishwashers must the compan produce and sell in order to maimize profit? 67) 68) A hotel has 230 units. All rooms are occupied when the hotel charges $110 per da for a room. For ever increase of dollars in the dail room rate, there are rooms vacant. Each occupied room costs $34 per da to service and maintain. What should the hotel charge per da in order to maimize dail profit? 68) 69) An outdoor sports compan sells 896 kaaks per ear. It costs $14 to store one kaak for a ear. Each reorder costs $8, plus an additional $9 for each kaak ordered. In what lot size should the store order kaaks in order to minimize inventor costs? 69) 70) A bookstore has an annual demand for 38,000 copies of a best-selling book. It costs $0.60 to store one cop for one ear, and it costs $55 to place an order. Find the optimum number of copies per order. 70) 71) A compan estimates that the dail revenue (in dollars) from the sale of cookies is given b R() = Currentl, the compan sells 900 cookies per da. Use marginal revenue to estimate the increase in revenue if the compan increases sales b one cookie per da. 71) 72) A grocer store estimates that the weekl profit (in dollars) from the production and sale of cases of soup is given b P() = and currentl 1300 cases are produced and sold per week. Use the marginal profit to estimate the increase in profit if the store prodcues and sells one additional case of soup per week. 72) 73) A compan estimates that the dail cost (in dollars) of producing chocolate bars is given b C() = Currentl, the compan produces 600 chocolate bars per da. Use marginal cost to estimate the increase in the dail cost if one additional chocolate bar is produced per da. 73) 74) The total cost, in dollars, to produce DVD plaers is C() = Find the marginal cost when = 3. 74) 75) The profit, in dollars, from the sale of compact disc plaers is P() = Find the marginal profit when = ) 6

7 76) Suppose that the dail cost, in dollars, of producing televisions is C() = , and currentl 60 televisions are produced dail. Use C(60) and the marginal cost to estimate the dail cost of increasing production to 63 televisions dail. Round to the nearest dollar. 76) 77) A suppl function for a certain product is given b S(p) = 0.08p 3 + 5p p + 4, where S(p) is the number of items produced when the price is p dollars. Use S (p) to estima how man more units a producer will suppl when the price changes from $13.00 per unit to $13.40 per unit. 77) 78) Suppose the demand for a certain item is given b D(p) = -4p2 + 3p + 8, where p represents the price of the item in dollars. Currentl the price of the item is $20. Us marginal demand to estimate the change in demand when the price is increased b one dollar. 78) 79) The volume of a sphere is given b the formula V = 4 3 πr3 where r is the radius. A tumor 79) is approimatel spherical in shape. Use V (r) to estimate the increase in volume of the tumor if its radius increases from 7 mm to 8 mm. Round to the nearest 100 mm 3. 7

8 Answer Ke Testname: HW2 1) Relative minimum at (2, 3) 2) Relative maimum at (-10, 81) 3) Relative maimum at - 1 6, ) ) Relative minimum at (6, -1.3) 5) Relative maimum at (0, 1); relative minimum at (2, -3) 6) Relative maimum at - 3 2, 79 ; relative minimum at 8 3, ) Relative minimum at (0, 32) 1 8) Relative maimum at 4, ) Relative maimum at (0, -4) 10) Relative minimum at -1, - 4 ; relative maimum at 1, 4 11) No relative etrema eist 12) Relative minimum at (-6, 6) 13) Relative minimum: (4, 1) 14) Relative maimum: (0, -1); relative minima: (3, -82), (-3, -82) 15) Relative maimum: (-9, 49) 16) No relative etrema eist 3 17) Relative maimum: 16, ) Relative minimum: (-3, 0) 19) Relative minimum: (-2, -64), relative maimum: (2, 64) 20) (0, 11), (250, 62,511), (500, 11) 21) (0, 98.6), (7, 103.5), (14, 98.6) 22) (150, 51.4) 23) 25) 26) ) Answers will var. 28) Answers will var. 29) Answers will var. -5 8

9 Answer Ke Testname: HW2 30) 31) 32) ) (0, 5) 34) (0, 3) 35) (0, 0) 36) (3, 4) 37) (0, 10), (2, 6) 38) Increasing on (-, -2], decreasing on [-2, ) 42) Concave up for all 43) Concave up on (0, ), concave down on (-, 0) 44) Concave up on (-, 0), concave down on (0, ) 45) Concave up on (-1, ), concave down on (-, -1) 46) Answers will var. 47) Answers will var. 48) Answers will var. 49) Answers will var. 50) $2,790,000 51) $863,333 52) (4.67, ) 53) (10, 13,000) 54) Absolute maimum: -21, absolute minimum: ) Absolute maimum: 14, absolute minimum: -4 56) Absolute maimum: 256, absolute minimum: ) Absolute maimum: , absolute minimum: ) Absolute maimum: 347, absolute minimum: ) No absolute maimum; absolute minimum: 3 60) 105 ft b 105 ft; 11,025 ft 2 61) 41.5 ft 41.5 ft 62) 9.11 ft 9.11 ft 63) 17 ft b 68 ft 64) 3.2 ft 65) 56 units 66) 3100 units 67) ) $187 69) 32 70) 2639 copies 71) $ ) $ ) $ ) $55 75) $149 76) $ ) 74 78) ) 600 mm 3 39) Increasing on -, - 1 4, decreasing on - 1 4, 40) Increasing on (-, ) 41) Decreasing on (-, -3] and [0, 3], increasing on [-3, 0] and [3, ) 9