課程名稱︰微積分甲上
課程性質︰必修
課程教師︰顏文明(陳宏代課)
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/07/14
考試時限（分鐘）：
試題 :
1. (20%) Find lim x->0 (x/(sinx)^2) - sinx/x^2
2. (20%) Evaluate lim n->∞(1/n)[ln((n+1)/n)/((n+1)/n)
+ ln((n+2)/n)/((n+2)/n)
+ .....
+ ln((n+n)/n)/((n+n)/n)]
3. (15%) Show that, for x > 0, 1/(1+x^2) < tan^-1(x) < x.
4. (10%) Evaluate ∫0 to 4 |x^(1/2) - 1|dx
5. (15%) Is it possible that the equation 3x = 2cosx + 5 has more than one
real root? Give reason to justify your answer. (You can draw the graphs
y = 3x and y = 2cosx + 5 to justify your claim.)
6. (0%) Find y' and y'' of the curve x^y = y^x
註: 超出範圍，本題不算分
7. (20%) Sketch the graph of f(x) = 1/(x^2 - 1) using the following guidelines
(a) Domain, (b) Intercepts, (c) Symmetry,
(d) Asymptotoes, (e) Intervals of Increase and Decrease
(f) Local Maximum and Local Minimun,
(g) Concavity and points of Inflection, (h) Sketch of the Curve