[試題] 97上 黃維信 微積分上 期末考
課程名稱︰微積分上
課程性質︰系必帶
課程教師︰黃維信
開課學院：工學院
開課系所︰工科海洋學系
考試日期（年月日）︰2009/1/15
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
lnx
1.Let f(x)= ── on (0,∞)
x
(a) find the intervals where f increases and the intervals where it decreases
(b) find the extreme values
(c)determine the concavity of the graph, inflection,and asymptotes (20%)
2.Calculate(40%)
dx
(a)5^(ln17)/(ln5) (e)∫──────
x^4-1
d sinxdx
(b)─[x^(2^x)] (f)∫──────
dx 1+(sinx)^2
(c)∫xarcsinxdx
dx
(d)∫────
√(a^2+x^2)
d 1
3. show that ─(arctanx)= ─── ,-1<x<1 (10%)
dx 1-x^2
4.Suppose that x=x(t),y=y(t)are twice differentiable functions that parameteri
-rize a curve.Take a point on the curve at which x'(t)≠0.
d^2y x'(t)y''(t)-y'(t)x''(t)
Show that──── = ───────────── if it exists. (10%)
dx^2 [x'(t)]^3
5.sketch the curve r=2sin3Θ in polar coordinate and find the area encloses
by the petal cruve.(15%)
6.Find the area of the surface generated by revolving the curve 6xy=x^4+3
from x=1 to x=3 about the x-axis.Determine the centroid of the volume enclose
by the surface,x=1 and x=3(15%)
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