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Unformatted text preview: AMS 510.01 Fall 2005, Final Exam Name: Student ID: Score: /200 The exam is in three parts: • The first section consists of problems involving the fundamental concepts of single and multiple variable calculus. Do all questions from this portion. (50 points) • The second section consists of comprehensive problems in Linear Algebra. Choose three out of the four problems to do. (75 points) • The third section consists of comprehensive problems in Advanced Calculus. Choose three out of the four problems to do. (75 points) Section 1. Fundamentals of Multivariate and Integral Calculus 1. Indicate whether each of the following statements is true or false. (2 points each) (a) The integral F ( x ) = R f ( x ) dx is unique for any given function f ( x ). (b) If two functions are equal everywhere in a closed interval ( f ( x ) = g ( x ) for all x in [ a, b ]), then their derivatives must be equal everywhere on the open interval ( a, b ). (c) A necessary and sufficient condition for the continuity of a function f ( x, y ) at the point ( x o , y o ) is that: lim x → x o ( lim y → y o f ( x, y )) = lim y → y o ( lim x → x o f ( x, y )) (d) R 1 R 1 xydxdy = 1. (e) ∂ 2 f ∂x∂y = ∂ 2 f ∂y∂x = for any function f ( x, y ) with continuous second order derivatives. 2. Consider the function f ( x ) =...
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 Fall '08
 Feinberg,E
 Calculus, Applied Mathematics, Derivative, lim

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