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Unformatted text preview: Part I Math 165 Final Exam (II) Spring 2003 1. Given f ( x ) = x +1 x- 1 x ≤ x- 1 x +1 x > Which of the following statements are true? I f is discontinuous at x =- 1 II f is discontinuous at x = 0 III f is discontinuous at x = 1 IV lim x →- f ( x ) =- 1 V lim x → 1 f ( x ) = 0 (a) only I, III, V (b) only I, II, III (c) only I, II, III, IV (d) only II, V (e) only IV, V (f) only V (g) none of the above 2. f ( x ) = √ x +1- 1 x , find lim x → f ( x ) . 3. (a) Find the critical points of the function f ( x,y ) = x 3 + y 3- 6 xy and classify each as a relative maximum, a relative minimum, or a saddle point. (b) repeat for f ( x,y ) = 4 x 3- 12 y 2 + 24 xy + 10 4. A manufacturer can produce blank videotape cassettes at a cost of $2 apiece. The cassettes have been selling for $4 apiece, and at this price, consumers have been buying 4,000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month. At what price should the manufacturer sell the cassettes to maximize profit? 5. A manufacturer finds that in producing x units per day (for < x < 100 ), three different costs are involved: a fixed cost of $1,200 per day in wages, a production cost of $0.20 per day for each unit produced, and an ordering cost of 100 x 2 dollars per day. Determine the level of production that results in minimal total cost per day. 6. For some f ( x ) , f ( x ) = 3 x 2- 2 x 3 . Find the relative extrema of f . 1 7. At a certain factory, the total cost of manufacturing q units during the daily production run is C ( q ) = 0 . 2 q 2 + q +900 dollars. From experience it has been determined that approximately q ( t ) = t 2 +100 t units are manufactured during the first t hours of a production run. Compute the rate at which the total manufacturing cost is changing with respect to time 1 hour after production commences.production commences....
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