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Unformatted text preview: Math 20A Second Midterm Exam. November 19, 2002 VERSION 1 Instructions: Fifty-ﬁve minutes. No books or notes; graphing calculators without symbolic manipulation programs are permitted. Do all 5 problems in your blue book. Show all work; unsubstantiated answers will not receive credit. Turn in your exam sheet with your blue book. 1. (20 points) Diﬀerentiate the following functions: (a) x3 ln x. (b) sin e7x . 2. (20 points) Use diﬀerentials to estimate the volume of paint required to cover the surface of a cube, with sides of length 10 inches, with a 0.025 inch thick coat of paint. 3. (20 points) If a sphere of ice melts so that its surface area decreases at a rate of 1 cm2 /min, ﬁnd the rate at which the diameter decreases when the diameter is 20 cm. (Recall that the surface area of a sphere of radius r is 4πr2 .) 4. (20 points) Let f be a continuous function on [1, 5], diﬀerentiable on (1, 5), with f (1) = 0 and f (x) ≥ 1 for 1 < x < 5. Find the smallest possible value of f (5). Justify your answer. 5. (40 points) Let f (x) = 3xe−x . (a) Find the local maxima and local minima of f , if any. (b) Find the intervals on which f is increasing and the intervals on which f is decreasing. (c) Find the inﬂection points of the graph of f . (d) Find the intervals on which the graph of f is concave up and the intervals on which the graph of f is concave down.
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This note was uploaded on 11/09/2010 for the course MATH 20A taught by Professor Staff during the Fall '08 term at UCSD.
- Fall '08