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Math 241 Name: Final Exam 1. [6 points] Consider the function f (x, y) = 1 4 3 x − 2xy + y 2 − y. 2 2 Determine whether the critical point (1, 1) for f is a local maximum, a local minimum, or neither. 2. [6 points] Evaluate Z 1Z 1 y e 0 x2 √ dy dx by changing the order of integration. y 3. [6 points] Find the equation of the tangent line to the curve x(t) = t 3 − 2t, t 3 + 1 at the point (−1, 2). (Your answer should be an equation involving x and y.) 4. [6 points] Find the equation of the tangent plane to the surface X(s,t) = 2t cos s, t sin s, t 3 at the point (4, 0, 8). 5. [24 points (4 pts each)] The following figure shows a contour plot for a function f : R2 → R. 5 0 1 2 3 4 6 7 5 8 20 4 5 4 10 18 16 3 12 14 14 3 16 2 2 6 4 2 12 14 8 10 18 1 0 1 0 1 2 3 4 5 6 7 0 (a) The function f has three critical points in the rectangle 0 ≤ x ≤ 7, 0 ≤ y ≤ 5. Estimate the coordinates of each of these points. (b) For each of the three critical points in part (a), indicate whether the Hessian H f at the point is positive definite, negative definite, or neither. (c) Estimate the maximum value of the function f (x, y) on the rectangle 0 ≤ x ≤ 7, 0 ≤ y ≤ 5. For the problems on this page, your answer must be correct to within 10% to receive full credit. 5 0 1 2 3 4 6 7 5 8 20 4 5 4 10 18 16 3 12 14 14 3 16 2 2 6 4 2 12 14 8 10 18 1 0 1 0 1 2 3 4 5 6 7 0 Z 2Z 2 f (x, y) dy dx. (d) Estimate 1 0 ZZ f (x, y) dA, where D is the disk of radius 1 centered at the point (3, 3). (e) Estimate D Z f (x, y) ds, where L is the line segment from the point (0, 2) to the point (3, 2). (f) Estimate L ZZ 6. [10 points] Evaluate S and (3, 1). y2 dA, where S is the square region in R2 with vertices (1, 0), (0, 2), (2, 3), Z 7. [10 points] Evaluate terclockwise. (−y, x) · ds, where C is the top half of the ellipse x2 + 4y2 = 4, oriented coun- C 8. [4 points] Find the coordinates of the point at which the helix x(t) = cos(πt), sin(πt),t intersects the paraboloid z = 4 − x2 − y2 . 9. [16 points)] Let S be the portion of the surface z = ln(r) in the range 0 ≤ z ≤ 1, with upward-pointing normal vectors. (a) Evaluate ZZ p x2 + y 2 dS. S ZZ (b) Evaluate S e z k · dS. 10. [12 points] Evaluate Z 1 Z √1−x2Z x2 +y2p −1 0 0 x2 + y2 dz dy dx by changing to cylindrical coordinates.