[試題] 97上 微積分乙 王振男 期中考
課程名稱︰微積分乙
課程性質︰系必帶
課程教師︰王振男
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰11/11
考試時限（分鐘）：150min
試題 :
1.Find the following limits:
x
x-∫cos(t^2)dt
0
(a)(5%) lim ──────────
x→0 6sin^(-1)x-6x-x^2
1
(b)(10%) lim sin(──)((㏑x)^4)*√(x)
x→+∞ x
2.(15%)Use the first derivative test to determine all local maxima and minima
of f(x) = x^3 - |x| on (-∞,∞)
3.(20%)Evaluate the following limit.
1 n k
lim ── Σ cos^(4)(──)
n→∞ n k=1 n
4.(15%)Let D be the region under the curve y = √(x) from 0 to 1.
Assume that the density function of D is ρ(y) = y. Find the center
of mass of D.
5.(10%)Find the equations for the tangent and normal to the cissoid of Diocles
y^2(2-x) = x^3 at (1,1)
6.(10%)Find the length of the curve
╭
｜x = ㏑(sect + tant) - sint
｜
｜y = cost, 0<=t<=π/3
╰
7.(15%) Paper folding： A rectangular sheet of 8.5-in.-by-11-in. paper is
placed on a flat surface. One of the corners is placed on the
opposite longer edge, as shown in the figure, and held there as the
paper is smoothed flat. The problem is to make the length of the
crease as small as possible. Call the length L. Try it with paper.
D C
┌────
｜ ︳
R｜\＼ ︳
｜ \ ＼ ︳
√(L^2-x^2)｜ L\ ＼︳Q (originally at A)
｜ \ /︳
｜ \/x︳
￣￣￣￣ B
A P
a.Show that L^2 = 2x^3/(2x-8.5)
b.What value of x minimizes L^2?
c.What is the minimum value of L?
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11/28 23:53, , 1F
11/28 23:53, 1F
※ ouanonym:轉錄至看板 NTUmed99 11/07 23:49
※ ouanonym:轉錄至某隱形看板 11/07 23:51