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Unformatted text preview: PRACTICE EXAM 1 103 (1) Compute 99 (2x - 198)2 x - 99 dx where here [x] is defined to be the largest integer x. (2) Let S be a square pyramid with base area r2 and height h. Using Cavalieri's Theorem, determine the volume of the pyramid. (3) Let f be an integrable function on [0, 1]. Prove that |f | is integrable on [0, 1]. (4) The well ordering principle states that every non-empty subset of the nat ural numbers has a least element. Prove the well ordering principle implies the principle of mathematical induction. (Hint: Let S P be a set such that 1 S and if k S then k + 1 S. Consider T = P - S. Show that T = .) (5) Suppose limxp+ f (x) = limxp- f (x) = A. Prove limxp f (x) = A. 1 MIT OpenCourseWare http://ocw.mit.edu 18.014 Calculus with Theory
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- Fall '10
- Calculus, Natural number, Cavalieri, base area r2, largest integer x.