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MATH 101 HOMEWORK 2 Due on Wednesday Sept. 17 1. (6 marks) Evaluate the following limits: n5 − 7n , n→∞ n2 + n3 − 3n5 (a) lim (b) lim sin(nπ). n→∞ 2. (6 marks) Evaluate: (a) n X 32n−i . i=1 i n X X (b) (i + j). i=1 j=1 (Hint: you may need the formulas in Theorem 1, Section 5.1.) R3 2 3. (4 marks) Write the upper and lower Riemann sums approximating 1 ex dx, corresponding to the partition of [1, 3] into n intervals of equal length. Do not attempt to evaluate these sums! 4. (4 marks) Prove that the area of the region enclosed by the lines y = 0, x = 1, x = 3, 2 and the graph of y = ex is at least e(1 + e3 ). In this problem, you are not allowed to use a calculator. 1