Exam 1 - Find parametric equations of the line tangent to...

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MATH 253.504–506 An unsupported answer Exam 1, version A is a wrong answer! 9/29/99 1. (10 pts.) Find the equation of the plane which contains the point (2 , 1 , - 1) and is perpendic- ular to the planes 3 x + y - z =6and2 x +2 y + z =4 . 2. (12 pts.) (a) Sketch the domain of f ( x, y )= 1 - x 2 + p 4 - y 2 . (b) What is the range of f ? Support your answer! 3. (10 pts.) Does lim ( x,y ) (0 , 0) xy x 2 + xy + y 2 exist? Why or why not? 4. (10 pts.) For what value(s) of the constant a ,ifany ,does u ( x, y )= e 2 x cos( ay ) satisfy Laplace’s equation 2 u ∂x 2 + 2 u ∂y 2 =0? 5. (10 pts.) Find the equation of the plane tangent to the surface x 2 + xy - yz 3 + z =6atthe point (2 , 1 , 1). 6. (12 pts.) (a) Define: f ( x, y )is differentiable at a point ( a, b ). (b) Use the definition to prove that f ( x, y )= xy is differentiable at every point ( a, b
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Unformatted text preview: Find parametric equations of the line tangent to the curve x = t 2-1, y = 2 t , z = t 2 + 1 at the point (3 ,-4 , 5). 8. (12 pts.) For f ( x, y ) = x 2 + 3 xy-y 2 , (a) What is the directional derivative of f at (1 , 2) in the direction from (1 , 2) to (4 , 6)? (b) For what unit vector ~u is the direction derivative D ~u f (1 , 2) largest? 9. (14 pts.) Suppose that z = f ( x, y ) where x = 2 u + 3 v and y = 3 u-2 v , and that f has continuous second partials. (a) Find z u in terms of the partial derivatives of f (in other words, in terms of z x and z y ). (b) Find 2 z u 2 in terms of the rst and/or second partial derivatives of f ....
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This note was uploaded on 03/26/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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