課程名稱︰微積分甲上
課程性質︰暑修
課程教師︰周青松
開課學院：
開課系所︰數學系
考試日期（年月日）︰2013/7/9
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1) Find F'(c) if it exists.
(a) f(x)=｛x^2 x≦1 ,c=1
｛2x-1 x>1
(b) f(x)=｛-x^2/2 x<3 ,c=3
｛-3x x≧3
(20%)
2) Find A and B given that the derivatives
of the following functions are everywhere continuous.
(a) f(x)=｛Ax^3+Bx+2 x≦2
｛Bx^2-A x>2
(b) f(x)=｛Ax^2+B x<-1
｛Bx^5+Ax+4 x≧-1
(20%)
3) Derive the following formulas.
(a) d p
─x^p/q=─x^p/q-1, for p/q a rational number.
dx q
(b) d p du
─u^p/q=─u^p/q-1─, for u a differentiable function of x.
dx q dx
(20%)
4) Find the intervals on which f increases
and the interval on which f decreases.
(a) f(x)=｛x^3 x<1
｛x/2 +2 x≧1
(b) f(x)=｛x+7 x<-3
｛|x+1| -3≦x<1
｛5-2x 1≦x
(20%)
5) Determine f on the domain indictaed given the following informations.
(a)(0,∞), f'(x)=x^(-5)-5x^(-1/5), f(1)=3.
(b)(-∞,∞),f'(x)=2+sinx, f(0)=3.
(20%)
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