作者ayabf (森~)
看板NTU-Exam
标题[试题] 97年_周青松_微积分甲上_期中考
时间Fri Nov 14 13:13:39 2008
课程名称︰ 微积分甲
课程性质︰ 数学 - 微积分
课程教师︰ 周青松
开课学院:
开课系所︰生工、地质、工管
考试日期(年月日)︰2008/11/14
考试时限(分钟):8:20~10:00
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
Ⅰ.A) Give necessary and sufficient conditions on A and B for the function
{Ax-B , x≦1
f(x) ={3x , 1<x<2
{Bx^2-A , 2≦x
to be continuous at x=1 but discontinuous at x=2.
B) Let f be the Dirichlet function
f(x)={ 1 , x rational
{ 0 , x irrational
Show that lim xf(x)=0
x→0
II.
A) Find A and B given that the function
f(x) = { x^2-2 , x <= 2
{ Bx^2+Ax, x > 2
is differentiable at x=2
B) Set f(x)={ x^2-x , x<=2 Show that f is continuous at x=2,
{ 2x-2 , x>2 is f differentiable at x=2?
III Let f be a differentiable function. Use the chain rule to show that:
A) If f is even , then f' is odd.
B) If f is odd , then f' is even.
IV A) Set f(x)=(secx)^2 and g(x)=(tanx)^2 on the interval(-π/2 , π/2)
Show that f'(x)=g'(x) for all x in (-π/2 , π/2).
B) Assume that f and g are differentiable on the interval (-c , c),
c>0 , and f(0)=g(0). Show that if f'(x)>g'(x) for all 属於(0,c),
then f(x)>g(x) for all x属於(0,c).
V Sketch the graph of f(x)=1/4‧x^4 - 2(x^2) + 7/4 ,x属於[ -3, 无穷大).
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