MATH 101 HOMEWORK 2
Due on Wednesday Sept. 22.
For full credit, show all work. Calculators are not allowed.
1. (6 marks) Evaluate the limits:
n + 2n2 − n3
,
n→∞
1 + 2n3
(a) lim
1 − cos(2/n)
.
n→∞ 1 − cos(3/n)
(b) lim
2. (6 marks) Evaluate the sums:
(a) 92 + 122 + 152 + 182 + . . . + (3n − 3)2 + (3n)2 ,
n X
i
X
(
2i+j ).
(b)
i=1 j=1
R5
3. (4 marks) Write the upper and lower Riemann sums approximating 2 x−3 dx, corresponding to the partition of [2, 5] into n intervals of equal length. (Do not try to evaluate
the sums.)
4. (4 marks) Identify the limit
lim
n→∞
n
X
3n − 2i
i=1
n2
as an area of a planar region, and use this to evaluate it.
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