final_review

# 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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## This note was uploaded on 10/21/2010 for the course MATHEMATIC stats taught by Professor Various during the Spring '10 term at Waterloo.

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final_review - Math 237 Final Review Problems 1...

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