M AT H 1 2 0 Differential and Integral Calculus Fa l l 2 0 1 6 - P r o b l e m S e t # 1 0 Due Wednesday, November 27. Write your name and student number very clearly in the upper right hand corner of your front sheet and staple your sheets if necessary. You are encouraged to collaborate with others when working on your homework assignments, but you must write up solutions independently, on your own. Do not copy the work of others. In accordance with academic integrity regulations, you must acknowledge in writing the assistance of any students, professors, books, calculators, or software. Be sure that your final write-up is clean and clear and effectively communicates your reasoning to the grader. 1) The upper right corner of a 8 12 rectangular piece of paper is folded down to the bottom edge as in the figure. How should the paper be folded so that the fold is the shortest? What is the length of the shortest possible fold? (Hint: Consider x and y as in the figure, and start by computing the cosine of the angles , , in the figure in terms of x and y.) 2) Review the -notation on page 372 of the textbook. Recall our definition of a Riemann sum. Given a function defined on the interval [ a, b] and given a partition a = x0 < x1 < x2 < . . . < xn1 < xn = b of a the interval [ a, b], with x = xi xi1 = b n , set n f (xi )x, Rn = i =1 where xi [ xi1 , xi ]. Now for x > 0 let f ( x ) = x 2 , a = 1, and b = 2. (a) Let xi = xi1 xi . Show that xi1 xi xi . (b) With the choice of xi as in (a), find Rn = n f (xi )x. i =1 (c) Use the expression for Rn in (b) to find 3) Consider the definite integral Z 2 1 1 t Z 2 1 f ( x )dx. dt. (a) By dividing the interval [1, 2] into n equal parts and choosing suitable sample points, show that n 1 n+i i =1 < Z 2 1 1 t n 1 dt < i =0 1 . n+i (b) How large should n be to make sure that any of the sums in part (a) approximates Z 2 1 1 t dt with an error of at most 5 108 ? (Hint: what is the difference between the overestimate and the underestimate in part (a)?)