課程名稱︰微積分甲上
課程性質︰暑修
課程教師︰周青松
開課學院：
開課系所︰數學系
考試日期（年月日）︰2013/7/3
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) Decide whether or not the indicated limits exist.
Evaluate the limits that do exit.
(a) lim (x^2+x-12)/(x-3)
x→3
(b) lim (f(x)-f(1))/(x-1), f(x)=x^3
x→1
(do not use differentiation)
(20%)
2) (a)Let f(x)=｛A^2*x^2, x≦2
｛(1-A)x, x＞2
Find the value for A such that f is continuous at x=2.
(b)Give the sufficient and necessary conditions on A and B
such that the function
f(x)=｛Ax-B, x≦1
｛3x, 1＜x＜2
｛Bx^2-A, 2≦x
is continuous at x=1 but discontinuous at x=2
(30%)
3) Let f(x)=sinx.
Evaluate lim(f(c+h)-f(c))/h at c=π/4
h→0
(do not use differentiation)
(20%)
4) Let f(x)=｛x^2-x, x≦2
｛2x-2, x＞2
(a)Show that f is continuous at x=2
(b)Prove or disprove: f is differentiable at x=2
(30%)
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