1. (15%) Set In=
(ln x)ndx, n ≥ 1.
(a) (6%) Find I1.
(b) (6%) Express In+1 in terms of In. (c) (3%) Use (b) to find I4.
2. (12%) (a) (6%) Find
x + 1 x2+ x + 1dx.
(b) (6%) Find
dx
ex(e2x− 1). 3. (12%) (a) (6%) Evaluate
1
2
0
x2√
1 − x2dx.
(b) (6%) Find d dx
x2 x
dt 1 + t5. 4. (12%) (a) (6%) Evaluate
sec3θdθ. You can use the formula for
sec θdθ if you know it.
(b) (6%) Find the arc length of the curve y = x2
2 + 1 from x = 0 to x = 2.
5. (10%) Find the volume of the solid obtained by rotating about the y-axis the region bounded by x = 0, x = 1, y = 0, and y =√
x2+ 1.
6. (15%)
(a) (8%) Write down the fourth degree Taylor polynomial of f (x) = sin x at x = 0. Also, write down its remainder provided by Taylor’s Theorem.
(b) (7%) Find the numerical value of sin 20◦accurate to within 10−4. Your answer can be expressed in terms of π. Don’t have to bother replacing π by 3.14 · · · . But the remainder must be estimated in details to prove the asserted accuracy of your numerical value.
7. (12%) (a) (6%) Find lim
x→∞x1x. (b) (6%) Find lim
x→0
ln(1 + x2) 1 − cos x .
8. (12%) A rod is being carried horizontally down a hallway of 1m wide with a right-angled turn.
What is the maximal length of the rod that can be carried around the corner?
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