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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? 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? 3 (15 pts.) Calculate the following limits: (a) lim
sin t2 . t0 t (b) lim+
x0 ln(x2 + 2x) . ln x 4 (25 pts.) What are the dimensions of the lightest opentop cylindrical can that will hold 1000 cubic cm of liquid? 2 5 (25 pts.) Let f (x) = ex  2ex  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). 3 ...
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 Spring '07
 Osserman
 Critical Point

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