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︰管院 地理 經濟
考試日期（年月日）︰2007/11/22
考試時限（分鐘）：2小時40分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
請將每一步驟表達清楚，不可只寫答案
1. Find the limit.
x + sinx + 2√x
lim -----------------
x→+∞ x + sinx
2. Find the limit.
lim x (csc√2x)^2
x→0+
3. The curve y = ax^2 + bx + c passes through (通過) the point (1,2) and
is tangent to the line y = x at the origin (原點). Find a,b,and c.
4. Find the equations of the tangent line (切線) and normal line (法線)
to the curve (曲線) x^2(x^2 + y^2) = y^2 at the point (√2 /2,√2 /2).
5. Determine whether the function f(x) = x|x| on the interval [-1,1] satisfies
the hypotheses (假設條件) of the Mean (平均) Value Theorem. If it does,find
the value or values of c that satisfy the equation
f(b)-f(a)
----------- = f'(c)
b - a
in the conclusion (結論) of the Mean Value Theorem, and if not, explain
why.
6. Show that the function
g(t) = √t + √(1+t) -4
has exactly (恰好) one zero in the interval (區間) (0,+∞).
7. Sketch the graph of the given function, making use of any suitable
information you can obtain from the function and its first and second
derivatives.
f(x) = x^4 - 12x^3 + 48x^2 - 64x
8. Determine all the extreme value (極值) of the function
f(x) = |x^3 - 9x|
on the interval (區間) [-3,3].
9. A 13-ft ladder (梯子) is leaning (倚靠) against a house when its base
starts to slide away (滑開). By the time the base is 12 ft from the
house, the base is moving at the ground changing then?
10. A rectangle (長方形) is to be inscribed (內接) in a semicircle (半圓)
of radius 2. What is the largest area the rectangle can have,and what
are its dimensions (長,寬)?
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