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MATH 2210 - Final Exam - 12/11/03 Note: Each problem is worth 10 points. For full credit, explain your work and justify your answers. Name: → → → (1) Let u = ha, bi be a vector. Find the projection P of u in the direction of h1, 1i in terms of a → → → → → and b. Let N = u − P : Show that N is perpendicular to P . 1 (2) Calculate the following limits: 2x2 y . (a) lim (x,y)→(0,0) x4 + y 3 (b) x3 − y 3 . (x,y)→(0,0) x2 + y 2 lim 2 (3) Find the plane passing through the point (0, 1, 0) and perpendicular to the two following planes: • The tangent plane to f (x, y) = sin(xy) + 3xy 2 − 2exy at (0, 1). • The plane with normal vector h1, 1, 1i passing through the point (1, 1, 1). 3 (4) Let f (x, y, z) = x2 + y 2 + z 2 , 9x x+y+z the composition g ◦ f at the point (1, 1, 1). and g(x, y) = (y − x, 2xy 2 ). Find the derivative of 4 (5) Let f (x, y) = xy 2 − 5x2 y. Find the maximum and minimum value of f on the region above the graph of y = x2 and below the line y = 4. 5 (6) Find the maximum volume of a rectangular box with no top that can be built with 1200 square feet of material. 6 (7) Calculate the double integral (1, 1). RR R y 2 dA where R is the triangle with vertices (2, 0), (0, 1), and 7 RRR (8) Set up the triple integral f (x, y, z) dV where K is the region below the graph of the K function 4 − x2 − y 2 and above the x, y-plane. 8 (9) Let C be the circle of radius 2 and center (0, 1, 0) in the y, z-plane. Show that Z Z xyz 2 2 4xy dx + e dx − y dx − 2x y dy + y dz − x dz = 2xy 2 dx + z dx + 2xy dy − 2x2 dy − z dy. C C 9 (10) Let f (x, y, z) be a function. Calculate div(grad(f )) in terms of f and its partial derivatives. 10 (Extra-credit) Let A be the 3 by 3 matrix 1 2 0 A = 0 2 1 . 0 0 −1 → → → (a) Find all the numbers a such that A v = a v for some non-zero vectors v . [Hint: There are three such numbers a] → → → (b) For each a that you found in part (a), find a non-zero vector v such that A v = a v . 11