課程名稱︰微積分甲上
課程性質︰必修
課程教師︰顏文明
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/06/30
考試時限（分鐘）：50
試題 :
1. If f(x) = (x-1)(x^2+x+1)/((2^x)(x^4+1)^(1/2)), find f'(1). (15 pts)
2. Find an equation of the tangent line to the curve y = x^2+x^3+x^4
at the point P(1, 3). (15 pts)
3. Given ε> 0, find a positive number δ(ε) such that
|x-1| < δ(ε) => |(x-1)(x-2)/((x-3)(x-4))| < ε (15 pts)
4. Find two constants a and b such that d/dx [(e^x)(acosx+bsinx)] = (e^x)sinx
(15 pts)
5. Let f(x) = (x+1)/((x+2)(x+3)), find f^(100) (x) (15 pts)
6. (a) Find d/dx sin^-1(x/3)
(b) Find d/dx (1/3)(tan^-1(x/3))
(c) Find d/dx ln(x+(x^2+9)^(1/2))
(d) Find d/dx cosx/(1+sinx)
(e) Find d/dx (x+x^(1/2))^(1/2)
(25 pts)