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Unformatted text preview: The following questions appeared on some prior final exams. These are only questions for (optional) extra practice. Disclaimer, these problems do not necessarily constitute a complete or comprehensive list of topics for material covered in this course. 1. Let c R and define A = { x I  x < c } . Completely prove that sup ( A ) = c . 2. Suppose c > 1 is a constant and  f ( x )   x  c for all x R . Use the  definition of the derivative to completely prove that f is differentiable at 0. 3. Assume functions f , g : R R are differentiable at a point a . Give an accurate  proof of the sum rule for derivatives: ( f + g ) ( a ) = f ( a ) + g ( a ). 4. Suppose that a function f only takes on integer values and lim x f ( x ) exists. Prove that there exists a K > 0 so that f ( x ) is constant on the interval ( K, ). 5. Suppose that f is differentiable at a . Give a complete proof of the fact that f is continous at a . 6. Suppose the function f is defined for all x [ a,b ] and that f is nonde creasing on [ a,b ]. (a) Let P = { t ,t 1 ,...,t n } be a partition of [ a,b ]. Write a formula for L ( f,P ) and U ( f,P ) in as simple terms as possible.) in as simple terms as possible....
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 Summer '11
 Katherine

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