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：
開課系所︰牙醫系 藥學系 醫技系 物治系 職治系 農化系 公衛系
考試日期（年月日）︰2018/11/20
考試時限（分鐘）：110
是否需發放獎勵金：是
試題 :
1.(20 points)
(a) Evaluate lim(x/sinx)^(1/(1-cosx)).
x→0+
(It might be helpful to use ln(u/v)=ln u-ln v, u>0, v>0, when doing
differentiation.)
sinx+xcosx+tan(ax)
(b) Find the value of a∈R such that lim ——————————=L∈R、
x→0 x^3
{0} has a non-zero finite limit. Then find the limit L.
2.(40 points) Let f(x)=(x^(1/3))(x+4). Answer the following questions by
filling each blank below. Show your work (computaions and reasoning)
in the space following. Write none in the blank if the object asked
does not exist. Each blank weights 5 points.
4(x+a)
(a) It is known that f'(x)=————. Find a=___________________________.
3x^(2/3)
4(x+b)
(b) It is known that f"(x)=————. Find b=___________________________.
9x^(5/3)
(c) f(x) is increasing on the interval(s):_____________________________.
(d) f(x) is concave downward on the interval(s):_______________________.
(e) f(x) has local maximal point(s) at (x,y) = ________________________.
and local minimal point(s) at (x,y) = _____________________________.
(f) f(x) has inflection point(s) at (x,y) = ___________________________.
(g) Sketch the graph of y=f(x).
3.(20 points) Find the dimensions of a right circular cylinder of maximum
volume that can be inscribed in a sphere of radius a>0. What is the
maximum volume?
4.(20 points)
(a) If the function y=f(x) is differentiable, it is known that for x>0,
x^y = x^f(x) = e^(f(x)ln(x)) is also differentiable. Find the
derivative dx^y/dx in terms of x,y and y'.
(b) Find the line normal to the curve given by {(x,y):tan(xy/2)+ln(x-3y)
=x^y-1} at (x,y)=(1,0).