Final-Fall 2006

# Final-Fall 2006 - AMS 510.01 Fall 2006, Final Exam Name:...

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AMS 510.01 Fall 2006, 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. (100 points) The second section consists of comprehensive problems in Linear Algebra. Choose two out of the three problems to do. (50 points) The third section consists of comprehensive problems in Advanced Calculus. Choose two out of the three problems to do. (50 points) Calculators may be used to simplify arithmetic, but advanced functions (including graph- ing, matrix manipulations, diﬀerentiation or integration) should not be used. Possibly useful trignometric relations sin( x ) = 2 sin( x 2 ) cos( x 2 ) cos( x ) = cos 2 ( x 2 ) - sin 2 ( x 2 ) d d x (sin - 1 ( x )) = 1 1 - x 2 ; | x | ≤ 1 d d x (cos - 1 ( x )) = - 1 1 - x 2 ; | x | ≤ 1 d d x (tan - 1 ( x )) = 1 1+ x 2 ; x R Good luck! 1

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Section 1. Fundamentals of Advanced Calculus (solve all problems) 1. Indicate whether each of the following statements is true or false. (2 points each) (a) If a function is continuous on a closed interval [ a, b ], then it is diﬀerentiable on the open interval ( a, b ). (b) R b a 1 x dx = ln b a . (c) The derivative g ( x ) = f 0 ( x ) of any function, f ( x ), is unique. (d) If lim x x o (lim y y o f ( x, y )) = lim y y o (lim x x o f ( x, y )), then f ( x, y ) must be con- tinuous at ( x o , y o ). (e)
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## This note was uploaded on 02/09/2010 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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Final-Fall 2006 - AMS 510.01 Fall 2006, Final Exam Name:...

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