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Unformatted text preview: Math 237 - Sample Midterms Answers Here are the answers (and a couple of solutions) to the sample midterms. Of course, on the test you are required to always give full solutions. SAMPLE MIDTERM 1 ANSWERS: 1. Short Answer Problems a) Let f : R 2 R . What is the definition of f being continuous at a point ( a, b )? Solution: f is continuous at ( a, b ) if lim ( x,y ) ( a,b ) f ( x, y ) = f ( 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? Solution: f x and f y are continuous. c) State Taylors Theorem with second degree remainder. Solution: Let f : R 2 R . If f C 2 in some neighborhood of ( a, b ) then for all points ( x, y ) in the neighborhood there exists a point ( c, d ) on the line segment joining ( a, b ) to ( x, y ) such that f ( x, y ) = f ( a, b ) + f x ( a, b )( x a ) + f y ( a, b )( y b ) + 1 2 f xx ( c, d )( x a ) 2 + f xy ( c, d )( x a )( y b ) + 1 2 f yy ( c, d )( y b ) 2 d) Let f : R 3 R . What is the formula for the linear approximation of f at the point ( a, b, c )? Solution: L ( a,b,c ) ( x, y, z ) = f ( a, b, c ) + f x ( a, b, c )( x a ) + f y ( a, b, c )( y b ) + f z ( a, b, c )( z c ). 2. Let f ( x, y ) = radicalbig | 1 x 2 y 2 | . a) What is the domain and range of f ? Solution: Domain is R 2 , range is z 0. b) Sketch the level curves and cross sections of z = f ( x, y ). Level Curves: Cross sections: Math 237 - Sample Midterms Answers 3. Prove that if f : R 2 R is differentiable at ( a, b ) then f is continuous at ( a, b ). Solution: See course notes. 4. Let g : R 2 R and let f ( x, y ) = g ( y 2 , xy ). Find 2 f xy . What assumptions do you need to make about g so that you can apply the chain rule? Solution: f yx = [ g uv ( y 2 , xy ) (2 y )+ g vv ( y 2 , xy ) x ] y + g v ( y 2 , xy ). Assuming g has continuous second partial derivatives. 5. Let f ( x, y ) = ln(2 x 3 y ). a) Find the linear approximation L (2 , 1) ( x, y ) of f ....
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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.
- Spring '08