midterm_f08_soln

# midterm_f08_soln - Math 237 Midterm Solutions Fall 2008 1...

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Unformatted text preview: Math 237 - Midterm Solutions Fall 2008 1. Short Answer Problems [1] 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 ). [1] 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. [2] c) State Taylor’s 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 [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 ). Math 237 - Midterm Solutions Fall 2008 2. Let f ( x, y ) = radicalbig | 1 − x 2 − y 2 | . [2] a) What is the domain and range of f ? Solution: Domain is R 2 , range is z ≥ 0. [3] b) Sketch the level curves and cross sections of z = f ( x, y ). Level Curves: k = radicalbig | 1 − x 2 − y 2 | x 2 + y 2 = 1 ± k 2 , k ≥ Cross sections: z = radicalbig | 1 − c 2 − y 2 | z = radicalbig | 1 − x 2 − d 2 | y 2 ± z 2 = 1 − c 2 , z ≥ x 2 ± z 2 = 1 − d 2 , z ≥ Math 237 - Midterm Solutions Fall 2008 [4] 3. Prove that if f : R 2 → R is differentiable at ( a, b ) then f is continuous at ( a, b )....
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midterm_f08_soln - Math 237 Midterm Solutions Fall 2008 1...

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