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Student number Name [SURNAME(S), Givenname(s)] MATH 101, Section 212 (CSP) Week 1: Marked Homework Assignment Due: Thu 2011 Jan 13 14:00 HOMEWORK SUBMITTED LATE WILL NOT BE MARKED 1. Let f (x) = x − x2 , 0 ≤ x ≤ 2, and divide the interval [0, 2] into n subintervals of equal width. (a) Find the Riemann sum Rn taking the sample points to be right endpoints of each subinterval [xi−1 , xi ]. (b) Find the Riemann sum Ln taking the sample points to be left endpoints of each subinterval [xi−1 , xi ]. (c) Find the Riemann sum Mn taking the sample points to be midpoints of each subinterval [xi−1 , xi ]. In each case (a), (b) and (c) find i) the expression for general n, ii) the value for n = 10, iii) the value for n = 100, and iv) the limiting value as n → ∞. For ii)–iv) give the answers both as exact fractions and as decimals accurate to four decimal places. 2. For each of the following limits, i) express the limit as a definite integral, and ii) sketch a region whose area is equal to the value of the limit. Do not evaluate the limits. m X n 1 X i +1 (a) lim 2 i n→∞ n n i=1 k k2 (b) lim cos 2 m→∞ m2 k=1 m ! 3. Evaluate the definite integral by interpreting it in terms of areas. (a) Z √ 2 −2 4− x2 dx (b) Z 3 −1 (3 − 2u) du (c) Z 10 0 |y − 5| dy 4. Without attempting to evaluate the integrals, prove the following: √ Z 1√ Z π/4 √ π 2 π (a) 1 ≤ 1 + t2 dt ≤ 2 (b) cos y dy ≤ ≤ 4 2 0 −π/4 (c) Z 6 4 1 dx ≤ x Z 6 4 1 dx 8−x (d) 2 ≤ 5 Z 2 0 1 dv ≤ 2 1 + v2