This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 opentop 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  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). 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 ...
View
Full
Document
This note was uploaded on 04/07/2008 for the course MATH 21A taught by Professor Osserman during the Fall '07 term at UC Davis.
 Fall '07
 Osserman
 Critical Point

Click to edit the document details