final_sp03 - Name TA Math 10C Final Examination Section...

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Name: Section: TA: Time: Math 10C. Final Examination June 13, 2003 Read each question carefully, and answer each question completely. Show all algebraic steps; no credit will be given for unsupported answers. 1. Find an equation for the plane that contains the line x = 3 + 2 t , y = t , z = 8 - t and is parallel to the plane 2 x + 4 y + 8 z = 17. # Score 1 2 3 4 5 6 7 8 Σ
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2. A certain differentiable function f ( x, y ) satisfies the following equations. f (0 , 0) = 1, f x (0 , 0) = - 2, f y (0 , 0) = 3. (a) Find an equation for the plane tangent to the surface z = f ( x, y ) at the point (0 , 0 , 1). (b) Find the linear approximation to f (0 . 1 , - 0 . 1).
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3. Let W ( s, t ) = F ( u ( s, t ) , v ( s, t )), where u (1 , 0) = 2, u s (1 , 0) = - 2, u t (1 , 0) = 6, v (1 , 0) = 3, v s (1 , 0) = 5, v t (1 , 0) = 4, F u (2 , 3) = - 1, and F v (2 , 3) = 10. (a) Find W s (1 , 0). (b) Find W t (1 , 0).
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4. A function f ( x, y ) has gradient f (1 , - 1) = - 4 , 3 . (a) Find the directional derivative of f at (1 , - 1) in the direction u = - 5 13 , 12 13 . (b) Find the maximum possible value of the directional derivative of f at (1 , - 1).
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5. Consider the function f ( x, y ) = 8 x 3 - 6 xy + y 3 .
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