[試題] 105上 李聰成 微積分乙上 期中考
課程名稱︰微積分乙上
課程性質︰共同必修
課程教師︰李聰成
開課學院：管院
開課系所︰
考試日期（年月日）︰2016/11/10
考試時限（分鐘）：160分鐘 (10:20-13:00)
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. Is there a number a such that
3x^2+ax+a+3
lim ──────
x→2 x^2+x-2
exists? If so, find the value of a and the value of the limit.
2.Evaluate the limit.
lim (√(4x^2+3x)-2x ) 註:根號是整個4x^2+3x
x→+∞
3.Express the limit as the derivate of some function f at some point a.
Then evaluate the limit.
cosθ-0.5
lim ──────
θ→2 θ-(π/3)
4.Show the the function f(x) = x|x| has an inflection point at (0,0) but
f"(0) does NOT exist.
5.For what values of a and b is the line 2x+y=b tangent to the parabola
y = ax^2 when x=2?
6.Sketch the graph of the given function, making use of any suitable
information you can obtain from the fuction and its first and second
derivatives.
2x^2
f(x) = ──────
x^2 - 1
7.Find the absolute maximum and absolute minimum values of
f(x) = 2cos x + sin 2x
on thee interval [0, π/2].
8.Use implict differentiation to find an equation of the tangent line (切線)
to the curve
x^2 + y^2 = ( 2x^2 +2y^2 -x )^2
at the point (0, 1/2).
9.Two cars start moving from the same point. One travels south at 30km/h
and the other travels west at 72km/h. At what rate is the distance between
the cars increasing two hours later?
10.Find the points on the ellipse (橢圓) 4x^2 + y^2 =4 that are farthest
away from (離最遠) the point (1,0).
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