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21bmt1bsolns2

21bmt1bsolns2 - Midterm version Ib Solutions Problem 1...

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Midterm version Ib Solutions Problem 1. Label the following statements as either true or false. a) If g is a function defined on [0 , 1] , then integraltext 1 0 g ( x ) dx exists. b) Suppose h is a differentiable function with derivative h . Then, integraltext b a h ( x ) dx = h ( b ) h ( a ) c) Σ 1 k = 3 k 2 = Σ 4 k =1 ( k 1) 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 9 Solution 1. a) False. Consider g ( x ) = braceleftbigg 1 if x is rational 0 if x is irrational . This function is defined everywhere, but has upper and lower sums which converge to different values. b) False. If h ( x ) is continuous, then the integral must exist. There are, however, functions that are differ- entiable, yet whose derivatives are not continuous. Consider h ( x ) = braceleftbigg x 2 sin( 1 x ) if x negationslash = 0 0 if x = 0 . The lim a 0 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.

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21bmt1bsolns2 - Midterm version Ib Solutions Problem 1...

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