Mathematics 53

Sample Exam

1. If f (x) = ax4 + bx3 − 2x2 , find the value of a and b so that (1, 1) is a point of inflection of f . 2. Let f (x) be continuous and differentiable. Suppose that f (15) = 88 and f 0 (x) ≤ 10. Find the least possible value of f (6) using MVT. 3. Given the following information about the graph of f , fill-in the table. x f (x) f 0 (x) f 00 (x) CONCLUSION x<0 (a) (b) decreasing; concave upward x=0 0 (c) (d) relative minimum 0<x<2 (e) (f) increasing; concave upward x=2 -16 DNE (g) not a rel. extremum, nor a POI 2<x<4 (h) (i) increasing; concave downward x=4 -32 (j) (k) relative maximum 4<x (l) (m) decreasing; concave downward 4. Given f (x) =

(x − 1)2 (x + 1)2

f 0 (x) =

4(x − 1) (x + 1)3

f 00 (x) =

8(2 − x) (x + 1)4

(a) Find the vertical and/or horizontal asymptotes of f . (b) Determine the intervals for which f is increasing/decreasing. (c) Find all relative extrema of f . (d) Determine the intervals for which the graph of f is concave upward/downward. (e) Find all points of inflection of f . (f) Sketch the graph of f . Label asymptotes (equation), and the coordinates of the relative extrema and points of inflection.

5. Find the area of the largest rectangle with sides parallel to the coordinate axes which can be inscribed in the area bounded by the two parabolas y = x2 − 16 and y = −x2 + 16. 6. The SummerSunshine Company produces and sells x number of units daily. The total price for x units sold is 14400 , and the total production cost is given by the function given by the price function p(x) = x3 + 125x − x 3 2x C(x) = + 375x. Find the number of daily production units maximizing the daily total profit of the 3 company. [total profit = total revenue - total cost]

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