周輕鬆考古題
Ⅰ.
A. Determine the discontinuities,if any of the following function:
┌ 2x+1 , x≦0
f(x)= │ 1 , 0<x≦1
└ x平方+1 , x>1
B.Show that
sinx 1-cosx
(a)lim ─── =1 (b)lim ──── = 0
x→0 x x→0 x
Ⅱ.
A.Find f'(1) give that
┌ x平方 , x≦1
f(x)= │
└ 2x-1 , x>1
B.Let
┌ x平方-x ,x≦2
f(x)= │
└ 2x-1 ,x>2
(a)Show that f is continuous at 2
(b)Find f'_(2) and f'+(2)
Ⅲ.
A.Two functions f and g are said to be inverse functions if
(f。g)(x)=x and (g。f)(x)=x
Let f and g be differentiable. Show that if f and g are inverse functions,
1
then f'(y)= ─── , where y=g(x) ,provided g'(x)≠0
g'(x)
d r r-1
B.Show that ── X = rX , r any rational number.
dx
Ⅳ. Let f be differentiable on (a,b) and continuous on 〔a,b〕.
A.Prove that if there is a constant M such that f'(x)≦M for all xε(a,b),
then f(b)≦ f(a)+M(b-a).
B.Prove that if there is a constant m such that f'(x)≧m for all xε(a,b),
then f(b)≧ f(a)+m(b-a).
Ⅴ.Sketch the grapf of the function
f(x)=2sin三次方x+3sinx, xε〔0,π〕.
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