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Unformatted text preview: Midterm version Ia Solutions Problem 1. Label the following statements as either true or false. a) Suppose h is a differentiable function with derivative h ′ . Then, integraltext b a h ′ ( x ) dx = h ( b ) − h ( a ) b) If g is a function defined on [0 , 1] , then integraltext 1 g ( x ) dx exists. c) Σ 4 k =1 ( k − 1) 2 = Σ − 1 k = − 3 k 2 d) integraltext b a f ( x ) · g ( x ) dx = integraltext b a f ( x ) dx · integraltext b a g ( x ) dx e) (3 3 )(3 3 ) = 3 6 Solution 1. a) False. If h ′ ( x ) is continuous, then the integral must exist. There are, however, functions that are differentiable, yet whose derivatives are not continuous. Consider h ( x ) = braceleftbigg x 2 sin( 1 x ) if x negationslash = 0 if x = 0 . The lim a → h ′ ( a ) does not exist, even though the derivative is defined everywhere. In fact, this derivative oscillates so much near 0, that it is not integrable. b) False. Consider g ( x ) = braceleftbigg 1 if x is rational if x is irrational . This function is defined everywhere, but has upper and lower sums which converge to different values....
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 Spring '08
 Vershynin
 Math, Calculus, Derivative, Fundamental Theorem Of Calculus, dx

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