[試題] 98上 黃維信 微積分 期中考
課程名稱︰微積分上
課程性質︰工科海洋大一必修
課程教師︰黃維信
開課學院：工學院
開課系所︰工科海洋系
考試日期（年月日）︰98.11.12
考試時限（分鐘）：110min
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Give an ε δ proof for lim √x=1 (10%)
x→1
2. True or False?(10%)Justify your answers.A function f is defined on the
interval[a,b]
(a)If f(a)>0 and f(b)<0 then there must exist at least one number c in
(a,b),for which f(c)=0
(b)If f is continuous on [a,b]with f(c)=0 for some number c in (a,b),
then f(a) and f(b) have opposite signs.
3. Let f be a differentiable function. Use the chain rule to show that:if
f is odd,then f' is even.(10%)
4. Find equations for the lines tangent and normal to the curve
ysin2x-xsiny=π/4 at the points (π/4,π/2).(10%)
5. Find a formula for all order of derivative of y=(a-bx)^n.(10%)
6. Let G(x)={xsin(1/x) x≠0 Sketch a figure that shows the general nature
{ 0 x=0
of the graph of G.Determine whether G is continuous or differentiable
at x=0?(20%)
x
7. Let F(x)=∫t(t-3)^2 dt. Find the critical points,the points of inflection,
0
if any,determine the concavity,and sketch the graph of F.(20%)
x t
8. Let f be everywhere continuous and set F(x)=∫[t∫f(u)du]dt.Find (a)F'(x)
0 1
(b)F'(1) (c)F''(x) (d) F''(1) (10%)
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