課程名稱︰微積分乙上
課程性質︰必修
課程教師︰張志中
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013/11/19
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (10%) Evaluate the limits.
n
x
(a) (5%) n為自然數, lim ----- = ?
x→∞ x
e
lnx
(b) (5%) p>0, lim ----- = ?
x→∞ p
x
2. (12%)
arctan(x/3) 1 9+x^2
Let h(x) = --------------- + ---ln(-------), x≠0.
x 6 x^2
b 2 c d
(a) Determine a,b,c,d 屬於實數 such that h'(x) = ax (9+x ) (arctan(x/3)) .
(b) Find the line tangent to the graph of y=h(x) at x=√3.
3. (30%) Evaluate the limits.
x 1
(a) (10%) lim (--- - ---) = ?
x→1 x-1 lnx
1
(b) (10%) lim (secx)^(---) = ?
x→0 x^2
2/3 1
(c) (10%) lim x * sin--- * cosx = ?
x→-∞ x
4. (10%) 3
x
Show that the equation x + e = 0 has exactly one real root.
5. (38%)
x^3-3x^2+7x+3
Answer the following questions and sketch the graph of y=f(x)=---------------,
3x^2
x≠0.
(a) Evaluate f'(x) = ?
(b) Evaluate f''(x) = ?
(c) f(x) is decreasing on the interval(s): ?
(d) f(x) is concave upward on the interval(s): ?
(e) f(x) has local maximum/maxima at (x,y) = (if any) ? ,
and local minimum/minima at (x,y) = (if any) ?
(f) f(x) has inflection point(s) at x = (if any) ?
(g) f(x) has asymptote(s) (if any) = ?
(h) Sketch the graph of y=f(x).
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