作者peter913227 (Chelsea魔獸)
看板NTU-Exam
標題[試題] 101上 李聰成 微積分乙上 期中考
時間Sat Nov 10 21:09:16 2012
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）︰2012年11月8日
考試時限（分鐘）：160分鐘
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,不可以只寫答案。
1. Find numbers a and b such that
√(ax+b)-2
lim ─────── = 1
x→1 √(x)
2. Evaluate the limit.
lim [√(3x^2+8x+6)-√(3x^2+3x+1)]
x→+∞
3. Find a function f and a number a such that
(2+h^6)-64
lim ─────── = f'(a)
h→0 h
and evaluate the limit.
4. 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.
x^2
f(x) = ─────
x-1
5. Find the equations of the tangent line(切線) and normal line(法線) to the
curve
√(x)
y = ─────
1+x^2
at the point (1,1/2).
6. Find the points on the curve
2(x^2+y^2)^2 = 25(x^2-y^2)
where the tangent is horizontal.
7. Show that the function f(x) = x|x| has an inflection point(反曲點) at (0,0)
but f''(0) does not exist.
8. Find the absolute maximum and absolute minimun valus of
f(x) = 2cos(x)+sin(2x)
on the interval (0,π/2).
9. A plane flying horizontally at an altitude(高度) of 2km and a speed of 800
km/h passes directly over a rader station. Find the rate at which the
distance from the plane to the station is increasing when it is 3km away
from the station.
10. If 1200cm2 of material is avilable to make a box with a square(正方形)
base and an open top, find the largest possible volume of the box.
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