课程名称︰微积分乙上
课程性质︰必修
课程教师︰张志中
开课学院：
开课系所︰牙医系 药学系 医技系 物治系 职治系 农化系 公卫系
考试日期（年月日）︰2018/11/20
考试时限（分钟）：110
是否需发放奖励金：是
试题 :
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).