課程名稱︰微積分乙上
課程性質︰必修
課程教師︰李聰成
開課學院：
開課系所︰
考試日期（年月日）︰2015.11.16
考試時限（分鐘）：160 分鐘
試題 :
請將每一步驟表達清楚，不可以只寫答案。
1. Evaluate the limit or show that it does not exist.
x^2+x
lim ───────
x→0 √(x^4+2x^2)
2.Evaluate the limit or show that it does not exist.
_ ___ ___
lim √x (√x+4 - √x+2)
x→+∞
3. Express the limit as the derivative of some function f at some point a.
Then evaluate the limit.
sin(3+x)^2－sin9
lim ────────
x→0 x
4. Sketch the graph oh the given function, making use of any suitable
information you can obtain from the function and its first and second
derivatives.
x^2+1
f(x)= ─────
3x^2-1
5. Find the absolute maximum and absolute minimum values of
f(x)=2x^3-3x^2-12x+5
on the interval [-2,4].
6. Use implicit differentiation to find an equation of the tangent line(切線)
to the curve
x^4=4(x^2-y^2)
_
at the point (1,√3/2).
7. Show that the function f(x)=x|x| has an inflection point at (0,0) but f"(0)
does not exist.
8. Find a value of c satisfying the conclusion of the Mean Value Theorem for
f(x)=x^3-3x^2+2x+2
on the interval [0,1].
9. A 10-foot ladder leans against the side of a building. If the top of the
ladder begins to slide down the wall at the rate of 2ft./sec, how fast is
the bottom of the ladder sliding away from the wall when the top of the
ladder is 8 feet off the ground?
10. A farmer has 1200m of fencing and wants to fence off a rectangular field
that borders a straight river. He needs no fence along the river. What are
the dimension (長,寬) of the field that has the largest area?