1. (15%)
(a) (5%) State the fundamental theorem of calculus.
(b) (10%) A function f (x) satisfies
∫
x 0
f (t)dt =∫
1 x
t2f (t)dt +x16 8 +
x18
9 +C for all x, where C is a constant. Find the function f (x) and the constant C.
2. (7%) Evaluate lim
x→∞
ln(1 + x) ln(1 + x2)
.
3. (7%) Evaluate lim
x→0
( 1 x2−
1 sin2x).
4. (14%)
(a) (7%) Derive the Taylor expansion of tan−1x centered at x = 0. (The expression of the n-th term for general n ≥ 0 is required.)
(b) (7%) Find lim
x→0
3 tan−1x − 3x + x3
3x5 .
5. (14%) Evaluate the following two indefinite integrals.
(a) (7%) ∫ x sin−1xdx.
(b) (7%) ∫
ln x x ln x + xdx.
6. (6%) Find the n-th term of the Taylor expansions of sin(x2)centered at x = 0 for general n ≥ 0. (You may use the Taylor expansions of cos x and sin x without deriving them.)
7. (7%) Find the n-th term of the Taylor expansion of sin2x centered at x = 0 for general n ≥ 0. (You may use the Taylor expansions of cos x and sin x without deriving them.)
8. (10%) Find the arc length of the curve y = ln sec x for 0 ≤ x ≤ π
4. (You can use any integral formulas you know without deriving them.)
9. (20%) Let Ω be the region bounded by y = 1
x(3 − x), x = 1, x = 2 and x-axis.
(a) (7%) Find the area of Ω.
(b) (7%) Find the volume of the solid obtained by rotating Ω about the x-axis.
(c) (6%) Find the volume of the solid obtained by rotating Ω about the y-axis.
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