课程名称︰微积分甲上
课程性质︰必修
课程教师︰颜文明(陈宏代课)
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/07/14
考试时限（分钟）：
试题 :
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
注: 超出范围，本题不算分
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