sample_mid_1

sample_mid_1 - D u f ( a, b ) of f at ( a, b ) in the...

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Math 237 - Sample Midterm 1 1. Short Answer Problems a) Let f : R 2 R . What is the definition of f being continuous at a point ( a, b )? b) Let f : R 2 R . What condition on f x and f y guarantees that the linear approximation of f is a good approximation? c) State Taylor’s Theorem with second degree remainder. d) Let f : R 3 R . What is the formula for the linear approximation of f at the point ( a, b, c )? 2. Let f ( x, y ) = p | 1 - x 2 - y 2 | . a) What is the domain and range of f ? b) Sketch the level curves and cross sections of z = f ( x, y ). 3. Prove that if f : R 2 R is differentiable at ( a, b ) then f is continuous at ( a, b ). 4. Let g : R 2 R and let f ( x, y ) = g ( y 2 , xy ). Find 2 f ∂x∂y . What assumptions do you need to make about g so that you can apply the chain rule? 5. Let f ( x, y ) = ln(2 x - 3 y ). a) Find the linear approximation L (2 , 1) ( x, y ) of f . b) Use Taylor’s Theorem to show that | R 1 , (2 , 1) ( x, y ) | ≤ 15 2 ± ( x - 2) 2 + ( y - 1) 2 ² for x 2, y 1. 6. Let f ( x, y ) = 2 x 2 + xy 3 . a) State the definition of the directional derivative
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Unformatted text preview: D u f ( a, b ) of f at ( a, b ) in the direction of the unit vector u . b) Find the rate of change of f at the point (1 , 2) in the direction of the vector (1 ,-3). c) In what direction from (-2 , 1) does f change most rapidly and what is the maximum rate of change. 7. Determine if each of the following limits exist. Evaluate the limits that exist. a) lim ( x,y ) (0 , 0) x 2-xy-y 2 x 2 + y 2 . b) lim ( x,y ) (0 , 0) x 2- | x | - | y | | x | + | y | . 8. Consider the function f ( x, y ) = x 4 / 3 y x 2 + y 2 , ( x, y ) 6 = (0 , 0) , ( x, y ) = (0 , 0) . a) Where is f dierentiable on its domain? b) Based on your answer in part a), what can you conclude about the continuity of both f x and f y at (0 , 0)?...
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This note was uploaded on 12/10/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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