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.
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