MIT18_014F10_pset2

MIT18_014F10_pset2 - If not well-deFned, provide a...

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± Unit 2: The Integral Pset 2 Due September 24 (4 points each) (1) page 57: 9de (You may use 9abc as you already proved those for recitation) (2) page 60: 6 (3) page 70: 7 (4) page 70: 11ac (5) Prove, using properties of the integral, that for a,b > 0 ² a 1 ² b 1 ² ab 1 dx + dx = dx. 1 x 1 x 1 x DeFne a function f ( w ) = w 1 dx , for w R + . Rewrite the equation above 1 x in terms of the function f . Give an example of a function that has the same property as the one displayed here by f . (6) Suppose we deFne b s ( x ) dx = ³ s k ( x k 1 x k ) 2 for a step function s ( x ) a with partition P = { x 0 ,x 1 ,...,x n } . Is this integral well-deFned? That is, will the value of the integral be independent of the choice of partition? (If well-deFned, prove it.
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Unformatted text preview: If not well-deFned, provide a counterexample.) Bonus: DeFne the function (where n is in the positive integers) f ( x ) = x : x = 1 1 2 n : x = n 2 1 Prove that f is integrable on [0 , 1] and that f ( x ) dx = 0. 1 MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT18_014F10_pset2 - If not well-deFned, provide a...

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