課程名稱︰微積分甲上
課程性質︰必帶
課程教師：容志輝
開課學院：電資學院、工學院
開課系所︰電機系、材料系
考試日期（年月日）︰2012.10.08
考試時限（分鐘）：50分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.
┌ 0 if x= 0
f(x)=│ , g(x) = xf(x)
│ 1
└ x sin ── otherwise
x
(a)Show that f and g are continous at 0.
(b)Show that f is not differentiable at 0.
(c)Show that g is differentiable at 0 and give g'(0)
(d)If h(x) and l(x)*h(x) are differentiable then l(x) is differentiable.
Prove or disprove (you can give me a counterexample.)
If not, give me a condition to let the l'(x) exist.
2.If f(x)≦g(x) for all x in some open interval containing c, expect possibly
at x = c itself, and the limits of f and g both exist as x approaches c.
Show that lim f(x) ≦ lim g(x)
x->c x->c
3.
π
ysin2x - xsiny = ─
4
Find the equation for the lines tangent and normal to the curve at the point
π π
( ─ , ─ )
4 2
4.
1 + s 1
y = ─── , s = t - ─ , t = √x
1 - s t
Find the dy/dx at x = 2
5.Given an ε,δ proof for the following statements.
lim│1 - 3x│ = 5
x->2
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