010.104-2008-2-mid - Honor Calculus II < V: Midterm...

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Unformatted text preview: Honor Calculus II < V: Midterm Exam (October 18, 13:00-15:00) s2: %V IU lǣ k k. ( 200) M K+ c a Z @\ \ 1. (25 pts) Let S be the surface in R3 defined by (x - 1)2 + 2(y - 2)2 + 3(z - 3)2 = 1. For P S, let TP denote the tangent plane of S at P . Prove that the curve { P S | TP is contained in a plane in R3 . 2. (25 pts) Show that if a C 2 function f satisfies f (tx, ty) = t2 f (x, y) for any (x, y) and any real number t R, then f (x, y) = 1 2 2f 2f 2f (0, 0) + 2xy (0, 0) + y 2 2 (0, 0) . x 2 2 x xy y (0, 0, 0) } 3. (30 pts) For the function f (x, y) = sin(x cos y) (a) find the local maximum points, local minimum points, and saddle points of f ; (b) find the third-degree Taylor polynomial of f at (0, 0). 4. (25 pts) Determine the minimum of xy + yz + zx on the surface x2 + y 2 - z 2 = 1. 5. (30 pts) Let F : R2 R2 be a linear map with detF (0, 0) = 2. (a) Find detF (1, 0). (b) For the function G(x, y) = (x2 , x2 - y 2 ), find det(F G) (1, 1). 6. (25 pts) Find the line integral X F ds of the vector field 2 F(x, y, z) = (2xex 2 +y 3 + z cos y, 3y 2 ex +y 3 - xz sin y, x cos y) along the curve X(t) = (cos t, sin t, t), 0 t 2. 7. (20 pts) For the function f (x, y, z) = x3 + y 3 + z 3 + 3xyz, (a) show that there exists a differentiable function g(x, y) defined on an neighborhood of (1, 1) such that g(1, 1) = 1 and f (x, y, g(x, y)) = 6. (b) Find grad g(1, 1). 8. (20 pts) Consider the function xy(x2 - y 2 ) , (x, y) = (0, 0) f (x, y) = x2 + y 2 0, (x, y) = (0, 0). (a) Show that f is continuous at (0, 0). (b) Determine whether f is differentiable at (0, 0). ...
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