課程名稱︰微積分甲上
課程性質︰必修
課程教師︰李白飛
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰100/11/11
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1 1 1
1. Evaluate lim ( ------- + ------- + ... + ------- )
_____ _____ _____
n->∞ √n^2+1 √n^2+2 √n^2+n
___ ___ _ ___
2. Evaluate lim √x^3 ( √x+1 - 2√x + √x-1 )
x->∞
3. Find the lines that are tangent to both the graphs of
y = x^2 and y = -x^2 + 4x - 10.
4. Let f(x) be a differentiable function on R(實數). Suppose that |f'(x)|<=1
for all 實數x. Show that there exists at most one value of x>1/2 such that
f(x)=x^2.
5. Show that the equation x^2 = x*sin(x) + cos(x)
has two and only two real roots.
6. Show that x > sin(x) for all x > 0, and cos(x) > 1 - x^2/2 for all x =/= 0.
3 _________
7. Find the asymptotes of the graph of y = √x^2 (x-1).
(x-1) (x-2)
8. Sketch the graph of y = -----------.
x^2
9. An empty water tank is in the shape of an inverted right circular cone
with depth 10 m and top radius 5 m. The water is poured into the tank at a
rate of 2 m^3/min. How fast is the water surface expanding after 9 minutes?
10. Find the shortest distance from a given point (0,b) on the y-axis to the
parabola x^2 = 4y. (The number b may have any real value.)
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