課程名稱︰微積分乙上
課程性質︰系必修
課程教師︰王振男
開課學院：醫學院
開課系所︰醫學系
考試日期（年月日）︰2012/11/27
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題：
1.Let f(x)＝(x^k)/(1＋x^k), x≧0, where k is a positive integer greater than 1.
（a）（5％）Determine where f(x) is increasing and where it is decreasing.
（b）（5％）Where is the function concave up and where is it concave down？
Find all inflection points of f(x).
（c）（5％）Find lim f(x) and decide whether f(x) has a horizontal asymptote.
x→∞
（d）（5％）Sketch the graph of f(x) together with its asymptotes and
inflection points（if they exist）.
2.（10％）A cable that hangs between two poles at x＝－ln2 and x＝ln2 takes
the shape of catenary, with equation y＝(1/2)[e^x＋e^(－x)].
Compute the length of the cable.
3.Denote the size of the a population by N(t), and assume that N(t) satisfies
dN/dt＝N(e^－aN)－N^2, where a is a positive constant.
（a）（5％）Show that the nontrival equilibrium N* satisfies e^(－aN*)＝N*.
（b）（5％）Assume now that the nontrival equilibrium N* is a function of
the parameter a. Use implicit differentiation to show that N* is a
decreasing function of a.
4.（10％）Find the equation of the tangent line to the graph of
e^[arcsin(xy)]＋ln(lnx)＝1 at (e,0).
5.Find the following limits.
（a）（5％） 1 1
lim (─ － ────).
x→0 x arctanx
（b）（5％）
lim x^(sinx).
x→0+
（c）（10％） x^2 sinu
∫ ──── du
0 u
lim ────────.
x→0 sinx
6.（15％）Use the mean value theorem to show that 1＋arcsinx＜e^(arcsinx) for
all x∈(0,1).（Hint：Consider f(x)＝e^(arcsinx)－1－arcsinx）.
7.（15％）Determine all local maxima and minima of f(x)＝[x^(1/3)](x^2－4) for
all x∈R.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.240.103