课程名称︰微积分乙下
课程性质︰必带
课程教师︰张树城
开课学院：生科院
开课系所︰生科系.生技系
考试日期（年月日）︰2017/4/27
考试时限（分钟）：110
试题 :
I.(10%) Compute the followings:
(i) Find the limit of the squence:
根号2, 根号(2+根号2), 根号(2+根号(2+根号2)),....
x^2*y
(ii) lim ─────
(x,y)→(0,0) x^4+2*y^2
II.(10%) Verify following infinite series coverge or diverge?
∞ n^n
(i) Σ ─────
n=1 n!
∞ 1
(i) Σ (-1)^n ───
n=1 根号n
III.(10%) Determine all x in which the following power series converge:
∞ x^n
(i) Σ ─────
n=1 n!
∞ x^n
(i) Σ (-1)^n ───
n=0 n+1
IV.(10%)
(i) Find the Maclaurin's series for the function f(x) = sin x at x=0.
1 1
(ii)Find lim (─── - ───)
x→0 sin x x
V.(10%) Find (z对x偏微分) and (z对y偏微分) at (0,0,0) for
x^3+z^2+ye^(xz)+zcosy=0
VI.(10%) Define xy(x^2-y^2)
────── (x,y)≠(0,0)
f(x,y) = x^2+y^2
0 (x,y)=(0,0)
Compute fxy(0,0)和fyz(0,0).
VII.(15%)
(i)Find the normal line and tangent plane to the surface x^2+y^2+z=9
at(1,2,4)
(ii)Use the linearization method to find the approximate of
(根号27)*(三次根号1021)
VIII.(10%)
Find the absolute maximum and minimum values of f(x,y)=x^2+3y^2+2y
over the unit disk D={(x,y):x^2+y^2 <=1 }
IX.(15%)Let f(x,y) have continuous partial derivatives in an open region D
in R^2 containing a point (a,b) where fx(a,b)=fy(a,b)=0.
Define F(t):=f(a+th,b+tk), 0<=t<=1
and Q(t):=[h^2*fxx(a+th,b+tk)+2hk*fxy(a+th,b+tk)+k^2*fyy(a+th,b+tk)],
(i)By using the chain rule to show that
d
─F(t)=h*fx(a+th,b+tk)+k*fy(a+th,b+tk)
dt
and
d^2
── F(t)=Q(t)
dt^2
(ii)By using the Taylor theorem to show that
f(a+h,b+k)=f(a,b)+1/2*Q(t)
from c between 0 and 1.