課程名稱︰微積分甲上
課程性質︰二組
課程教師︰莊正良
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2009/11/06
考試時限（分鐘）：60 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.Find lim 1-cosx
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x→0 |x+1|-|3xsinx-x-1|
2.It is known that f(x)=2x+sinx is increasing, since f'(x)>=0 always. Let
f^(-1)(x)denote its inverse function. Find f^(-1)(2π) and f^(-1)'(2π)
3.Find y' and y'' of x^2-4xy+y^3+2=0
4.Let f(x)=ln(x)/x, x>0. Answer the following and show your work.
(a)f(x) is increasing in the intervals ____ and is decreasing in the
intervals ____.
(b)Local maxima occur(s) at x=__ and local minima occur(s) at x=__.
(c)Which of e^π and π^e is larger?
5.Let f(x)=9x^(2/3)(x^2/40+1/2). The graph of f(x) has inflection point(s) at
x=__, concaves upward in the interval(s) ___ and concaves downward in the
interval(s) ___. Show your work.
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