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Unformatted text preview: Math 20C Second Midterm Solutions (Version #2) Problem 1. Consider the following contour map of a function f , with its contour lines labeled by the corresponding function values. A B C E D 16 16 16 20 20 20 24 28 12 12 16 16 20 20 24 24 12 (a) The function f has critical points at A , B , and C . For each of these points, say whether it is a local maximum, a local minimum, or a saddle point. A and B are local maxima, since they are in the center of concentric contour lines and any direction you go from them is down. C is a saddle point, since in two directions (towards B and away from B ) you can go up from it, but in at least one other direction (down and left) you can go down from it. (b) Does the gradient vector at D point towards A , towards B , or towards E ? Towards A , since that is the direction of greatest increase in the function value (the direction in which one goes up the steepest). (c) The value of f at point E is 18 . Estimate the numerical values of f x and f y at E , to within 1 accuracy. One unit to the left of E , at (1 ,- 1), the function value is about 20, and one unit to the right, at (3 ,- 1), the function value is about 16, while we are given that at E = (2 ,- 1) 1 the function value is 18. Any two of these points give the same slope for the functionthe function value is 18....
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- Fall '07