t2 - tires in storage amount to $4 per year for one tire(a...

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Methods of Calculus MAC2233 Test 2 October 12, 1990 (1) (15 pts.) Sketch the graph of a function that has the properties: ( - 3 , - 2) and (1 , 4) are on the graph; f 0 ( - 3) = 0 and f 0 (1) = 0; f 00 ( x ) > 0 for x < - 1; f 00 ( - 1) = 0; and f 00 ( x ) < 0 for x > - 1. (2) (25 pts.) Sketch the graph of the function y = 2 x 3 - 6 x 2 - 48 x + 5, indicating all relative extreme points and inflection points. (3) (15 pts.) Consider the problem of finding the dimensions of the rectangular garden of area 111 square feet for which the amount of fencing needed to surround the garden is as small as possible. (a) Draw a picture of a rectangle with width w and length l . (b) Express the quantity to be minimized as a function of w . (4) (15 pts.) The Great American Tire Company expects to sell 600,000 tires of a particular size and grade during the next year. Sales tend to be roughly the same from day to day. Setting up each production run costs the company $20,000. Carrying costs, based on the average number of
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Unformatted text preview: tires in storage, amount to $4 per year for one tire. (a) Determine the costs incurred if there are 20 production runs during the year. (b) Express the total inventory cost as a function of the production run size. (5) (15 pts.) Some years ago, it was estimated that the total demand for steel approximately satisfied the demand equation p = 88-20 x , and the total cost of producing x units of steel was C ( x ) = 175+8 x . (The quantity x was measured in millions of tons and the price and total cost were measured in millions of dollars.) Determine the level of production and the corresponding price that maximizes the profits. (6) (15 pts.) Calculate the derivatives of the following functions. (a) f ( x ) = ± x x 2 + 1 ¶ 2 (b) g ( x ) = 2( x 3-1)(3 x 2 + 1) 4 (c) h ( x ) = 4 x 2 + x √ x 1...
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