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Unformatted text preview: Homework 7 Math CS 120, Winter 2009 Due on Thursday, February 26th, 2009 1. Let f : [ a, b ] R be a continuous function, differentiable in ( a, b ). Prove that if f ( x ) = 0 for all x ( a, b ), then f is a constant. 2. Let f : [ a, b ] R be a continuous function, differentiable in ( a, b ). Prove that if f ( x ) > 0 ( f ( x ) < 0), where x ( a, b ), there exists > 0 such that the function f is strictly increasing (decreasing) in the interval ( x , x + ). 3. Let f : [ a, b ] R be a continuous function, and let x [ a, b ]. Define F ( x ) = Z x x f ( t ) dt, x [ a, b ] . Prove that F : [ a, b ] R is continuous. 4. Prove the Fundamental Theorem of Calculus : Let f : [ a, b ] R be a continuous function, and let x [ a, b ]. Define F ( x ) = Z x x f ( t ) dt, x [ a, b ] . Prove that F is differentiable in ( a, b ), and that F ( x ) = f ( x ) for all x ( a, b ). As a consequence, show that f ( x ) f ( a ) = Z x a f ( t ) dt....
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 Fall '09
 Math

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