010-104-2002-2-mid - > 5 ; o 2 2002 10 26{ 1r 3r 4 Z 9...

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Unformatted text preview: > 5 ; o 2 2002 10 26{ 1r 3r 4 Z 9 < : s2: j h ]_ \ s&` "r r(8& 100&). HH x h 1. (10&) Consider the region bounded by the curve y = x + 1, the x h axis, and the y-axis. Let Vx , Vy be the volumes of the solid generated by revolving the region about the x-axis and the y-axis, respectively. Find the ratio Vx : Vy (Write it as simple as possible). (15&) h x2 y 3 , (x, y) = (0, 0) f (x, y) = x2 + 4y 3 0, (x, y) = (0, 0). (a) Find fx (0, 0) and fy (0, 0). (b) Prove that f is not continuous at the origin. 2. 3. 4. 5. (10&) Find the maxima, minima, and saddle points of f (x, y). h f (x, y) = sin x + sin y + sin(x + y). (10&) Find the points on the curve xy + 2xz = 5 5 nearest to the origin. h (10&) Let (0, 0) be a critical point of a differentiable function z = f (x, y). h Does the condition "fxx > 0 and fyy > 0" imply that f has a local minimum at (0, 0)? If not, find a counter example and a condition which f should satisfy to have a local minimum at (0, 0). (10&) Show that if the gradient vector of f (X) is parallel with X, then h f (X) is constant on every sphere centered at the origin. (May assume X = (x, y, z)) (10&) By using Taylor's formula, find the quadratic approximation of f h ex + e-x at (0, 0), where f (x, y) = ecosh x (sin y + 1). (Hint : cosh x = ) 2 (15&) Let z = f (x, y), x = g(z, y) be functions such that f (g(z, y), y) = z. h Show that 2 1 + gy 2 2 fx + fy = . 2 gz (10&) Compute h 0 1 16 1 2 1 y4 6. 7. 8. 9. cos(16x5 )dxdy. ...
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This note was uploaded on 09/09/2009 for the course MATHMATICS 010-102 taught by Professor Cho during the Spring '09 term at Seoul National.

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