课程名称︰微积分甲上
课程性质︰大一必修
课程教师︰容志辉
开课学院：工学院
开课系所︰材料系、电机系
考试日期（年月日）︰2014/10/13
考试时限（分钟）：50分钟
试题 :
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Calculus A-02 Quiz 1
sin x cot x - 1
1. Use the fact lim ──── = 1 to find lim ────── .
x→0 x x→0 1 / x
2. Suppose that the position function of some man is f(t) = t^2 - t + 10.
Which time t between t = 0 and t = 10 can imply the instantaneous velocity
equals the average velocity?
3. If f is continuous on [0,1] and 0 ≦ f(x) ≦ 1 for all x in [0,1], then
show that there exists a c in [0,1] such that f(c) = c.
4. Let x^2 y + x y^2 - 3x + 2y = 1.
(a) Show that y can be a function of x ( i.e., y can be expressed by x ) on
some neighborhood of the point (1,1).
(b) Find y'(x) and y''(x) at the point (1,1).
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