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final-practice-sols

final-practice-sols - Math 21A 1(15 pts Brian Osserman...

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Unformatted text preview: Math 21A 1 (15 pts.) Brian Osserman Practice Exam 3 Suppose f (x) is a twice dierentiable function on (a, b). Is it always true that every critical point for f (x) is a local extremum? (Justify your answer) What are two methods of testing whether a given critical point is a local max? Answer: No, not every critical point is a local extremum. For instance, f (x) = x3 has a critical point at x = 0, but not a local max or local min. One can check for a local max at a critical point by testing whether the sign of the derivative switches from positive to negative at that point, or by testing whether the second derivative is negative at that point. 2 (20 pts.) A spherical balloon is inated at the rate of 100 cubic feet per minute. How fast is the balloon radius increasing at the instant the radius is 5 feet? How fast is the surface area increasing? Answer: Let r be the radius in feet, t time in minutes, V volume in cubic dr dt feet, and A surface area in square feet. Then we get dA dt = 1 ft/min and = 40 square feet per minute. 3 (15 pts.) Calculate the following limits: (a) lim sin t2 . t0 t Answer: Using L'Hopital's rule gives 0. 2 (b) lim+ x0 ln(x + 2x) . ln x Answer: Using L'Hopital's rule gives 1. 4 (25 pts.) What are the dimensions of the lightest open-top cylindrical can that will hold 1000 cubic cm of liquid? Answer: The minimum weight (equivalently, minimum surface area) occurs when radius = height = 10/ 3 . 5 (25 pts.) Let f (x) = ex - 2e-x - 3x. Find the critical points, and the intervals on which f (x) is increasing and decreasing. Find the points of inection, and the intervals on which the graph is concave up and concave down. Sketch the graph of f (x). Answer: Critical points are at 0 and ln 2. f (x) is increasing on (-, 0) and (ln 2, ) and decreasing on (0, ln 2). There is a point of inection at x = ln 2 2 , and the graph is concave up on ( ln 2 , ) and concave down on 2 (-, ln 2 ). 2 (Graph omitted) 2 ...
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