課程名稱︰微積分甲
課程性質︰共同必修
課程教師︰周青松
開課系所︰
考試時間︰2006.04.21 08:10~09:50
試題 :
Ⅰ. Calculate the indicated limit.
1 x t^2
A) lim (── ∫ e dt)
x→∞ x 0
1 x 1
B) lim ── ∫ sin(────) dt
x→∞ x 0 t + 1
Ⅱ. Evaluate the improper integrals that converge.
∞ -px
A) ∫ e dx, p > 0
0
0 x
B) ∫ xe dx
-∞
∞
C) ∫ cosh x dx
0
π/2 cos x
D) ∫ ───── dx
0 √(sin x)
Ⅲ. A) Show that k
∞ (-1) 2k
cos x = Σ ──── x for all real.
(2k)!
B) Derive the series expansion:
3 5
1 + x x x
ln (────) = 2 ( x + ── + ── + … ) for -1 < x < 1
1 - x 3 5
Ⅳ. Evaluate the given limit in two ways: (a) Using L'Hôpital's rule,
and (b) Using power series.
sin x - x
A) lim ──────
x→0 x^2
x
e - 1 - x
B) lim ──────
x→0 x arctan x
Ⅴ. Find a power series representation for the improper integral.
x ln (1+t)
A) ∫ ───── dt
0 t
x sinh t
B) ∫ ──── dt
0 t
（每大題均20分）
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