final_review - Math 237 Final Review Problems 1....

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Math 237 Final Review Problems 1. Domain/Level Curves/Cross-sections 2. Limits/Continuity/Differentiability a) Write the precise definition of the limit. Solution: We write lim ( x,y ) ( a,b ) f ( x,y ) = L if for every ± > 0 there exists a δ > 0 such that | f ( x,y ) - L | < ± whenever 0 < k ( x,y ) - ( a,b ) k < δ . b) How do you try to evaluate a limit? Solution: First, see if we can observe the function is continuous. Otherewise, try to prove it doesn’t exist by approaching along different lines and curves. If we get two limits with different values then the limit doesn’t exist. If all lines and curves give the same value L then we try to prove the limit exists using the squeeze theorem. If we can not make the squeeze theorem work, then we must find two curves giving different values to prove the limit doesn’t exist. c) What is the definition of continuous. Solution: f ( x,y ) is continuous at ( a,b ) if lim ( x,y ) ( a,b ) f ( x,y ) = f ( a,b ). d) What is the definition of differentiability. Solution: f ( x,y ) is differentiable at ( a,b ) if 1. f x ( a,b ) and f y ( a,b ) both exist. 2. lim ( x,y ) ( a,b ) | f ( x,y ) - L ( a,b ) ( x,y ) | k ( x,y ) - ( a,b ) k = 0. e) What are two important theorems about differentiablity. Solution: If f x and f y are both continuous at ( a,b ) then f is differentiable at ( a,b ). If f is differentiable at ( a,b ) then f is continuous at ( a,b ). 3.
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final_review - Math 237 Final Review Problems 1....

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