課程名稱︰微積分甲上
課程性質︰大一必修
課程教師︰容志輝
開課學院：工學院
開課系所︰電機系、材料系
考試日期（年月日）︰2015/01/06
考試時限（分鐘）：50分鐘
試題 :
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CALCULUS I QUIZ 4, 2014 FALL
(Total 150 points)
1. ( 15% + 15% + 20% ) For (c), you should plot both curves.
(a) Show that the curve r = 2015 sinθ is a circle.
(b) For the curve x = -sin t, y = t + cos t where 0 ≦ t ≦ π/ 2, find
its arc length.
2
(c) Find the area of the region that lies inside r = 8 sin 2θ and outside
r = 2.
2. ( 20% + 20% ) Solve the equations.
2x 4x
dy -4e y + e - 1
(a) ── = ───────── , x > 0
dx 4x
e - 1
dy 2 2
(b) ── + y = ─── , x > 0 ( Hint: 考慮變數變換 )
dx 2
x
3. ( 10% + 20% ) This problem is about solids of finite volume but infinite
surface area. The classical one is obtained by rotating the curve y = 1 / x,
x ≧ 1 about the x-axis.
(a) Consider the integral
1 1
∫ ── dx.
0 p
x
For what p does the integral converge?
(b) Based on (a), construct a function h on (0,1] such that the volume of
2
the solid obtained by rotating h about the line segment { (x,0) ∈ R |
x ∈ (0,1] } on the x-axis is finite while the surface area is infinite
b 2
and verify your results. [Note: Volume = ∫ π( h(x) ) dx.]
a
4. ( 10% + 10% + 10% ) Consider the time-independent Schrodinger equation
2 2
h d ψ(x)
- ── ──── + Vψ(x) = Eψ(x) (1)
2m 2
dx
where h, m, E are constants. A solution ψ(x) to (1) describes a particle
b 2
and is called a wave function. Moreover ∫ |ψ(x)| dx represents the
a
probability of finding the particle between a and b.
(a) Find a non-zero, that is not y = 0, solution to (1) for the case V = 0.
[Note: Need not to find the general solution.]
(b) A particle is described by the wave function
x
A √( ─────── )
4 4 if 0 ≦ x ≦ L
ψ(x) = 1 + x / L
0 otherwise.
Find the expression of A > 0 such that the probability of finding the
particle between 0 and L is 1.
(c) Happy New Year~
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