1. (10%) Let F (x) =∫
x2 0
cos(t2+t) dt.
(a) (6%) Compute F′(x).
(b) (4%) Compute the limit lim
x→0
F (x) x2 . 2. (14%) Compute the integrals.
(a) (7%) ∫ sin
√
2x + 1 dx (b) (7%) ∫ tan−1( 2 x)dx 3. (18%) Compute the integrals.
(a) (9%) ∫ 1 x2
√
1 + x2dx (b) (9%) ∫
x2+4x − 1 x4−1 dx 4. (18%) For a constant a > 0, Ωa is the region bounded by y = x
x + 1, x = a and the x-axis. Σa is the region bounded by y = x
x + 1, y = a
a + 1 and the y-axis. (Ωa and Σa together make a rectangle as shown in the Figure.) Rotate Ωa
about the x-axis and let the resulting volume be U (a). Rotate Σa about the y-axis and let the resulting volume be V (a).
b )b!b!,!2b *
͈
͂
b b
y z
,
(a) (8%) Find U (a).
(b) (8%) Find V (a).
(c) (2%) Find U (a) − V (a).
5. (12%) Find the following limits.
(a) (6%) lim
x→0
cos x2−1
sin x4 (b) (6%) lim
x→1( 2 x2−1−
1 ln x) 6. (a) (4%) Write down the Taylor series of −1
√
1 − x2 at x = 0.
(b) (6%) Find the Taylor series of cos−1x at x = 0.
(c) (4%) Write down the 10th-degree Taylor polynomial, P10(x), of cos−1(x2)at x = 0.
(Note that you need to compute the exact values of coefficients for part (c). Do not express coefficients in terms of Cαn only.)
7. (14%) Suppose that f (x) = 1 2x2+
1
3 ⋅ 2x3+ ⋯ + 1
n(n − 1)xn+ ⋯, for ∣x∣ < 1.
(a) (2%) Find f(10)(0)
(b) (6%) Find the Taylor series of f′(x) and f′′(x) at x = 0. Recognize them as elementary functions.
(c) (6%) Express f (x) as an elementary function.
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