課程名稱︰微積分二
課程性質︰數學系大一必修
課程教師︰余正道老師
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2019/4/18
考試時限（分鐘）：180
(There are 120 points in total)
試題 :
1.[10%] Prove the mean value theorem: Suppose f(x,y) is a C^1 function defined
on an open disk centered at 0. Then for (ξ,η) in the disk, there
exists 0<t<1 such that f(ξ,η)-f(0)=fx(tξ,tη)ξ+fy(tξ,tη).
2.[10%] Consider the function sin(xy)
f(x,y)=————— on R^2\0
√x^2+y^2
(a) Show that lim f(x,y) exists
(x,y)->0
(b) After defining f(0,0) to be the limit in (a), is f(x,y) differentiable
at (0,0)? Justify your answer.
3.[10%] Let f(x,y) be differentiable at the point P0=(x0,y0). Consider the
tangent plane T of the surface S: z=f(x,y) at P0.
(a) Show that (the equation of) the tangent plane T is the limit of the
plane passing through the three points (xi,yi,zi), i=0,1,2, of S
where P1=(x1,y1) and P2=(x2,y2) approach P0 from distinct direction,
making an angle not equal to 0 or π.
(b) Let ε>0 and ξ(t), η(t) differentiable on (-ε,ε) such that
(ξ(0),η(0))=P0. Show that the tangent line of the curve
(ξ(t),η(t),f(ξ,η)) at t=0 lies on T.
4.[10%] Compute the line integral ∫ xdy-ydx
T﹣﹣﹣﹣﹣ where T is the rectangle with
x^2+y^2
vertices (±1,0) and (0,±1), oriented counterclockwise
5.[20%] Let S be a compact set in R^2.
(a) Let (Pk)∞ be a sequence in S. Show that there exists a subsequence of
k=1
Pk which converges to a point in S.
(b) Let f be a continuous function on S. Show that f is uniformly continuous.
6.[20%] Suppose f(x,y) is a C^1 function on a rectangle [a,b]x[c,d].
(a) Show that F(x):=∫d f(x,y) dy is differentiable and F'(x)=∫d fx(x,y) dy
c c
(b) Show that ∫b∫d f(x,y) dydx=∫d∫b f(x,y) dxdy
a c c a
7.[20%] Let f(x,y) be a C^1 function such that F(a,b)=0 and Fy(a,b)≠0.
The implicit function says that F(x,y)=0 is equivalent to y=f(x)
for a C^1 function f in a neighborhood of (a,b).
(a) Let n≧2 and suppose F(x,y) is a C^n function. Show that f(x) is a
C^n function.
(b) Let G(x,y) be a C^1 function such that it achieves a local maximal
at (a,b) subject to the condition F(x,y)=0. Prove that there exists
one and only one real number λ such that
(Gx(a,b),Gy(a,b))=λ(Fx(a,b),Fy(a,b)).
8.[20%] Suppose P is a critical point of a C^3 function f(x,y). Let H(x,y)
fxx fxy
=(fyx fyy) (matrix)
(a) Suppose detH(P)>0. Show that f has a local extremal at P.
(b) Suppose f(P)=0 and let detH(P)<0. Show that in a neighborhood of P,
the solutions f(x,y)=0 form two curves (x,y)=(ξi(t),ηi(t)),i=1,2,
where ξi,ηi are some C^1 functions. Compute the angle of these two
curves at P in terms of second derivatives of f.