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(MATH301)2001-2008_f_midterm_MATH301_by_cs_xsx420.pdf
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Notations: R denotes the set of all real numbers. When n appears in the subscript of a sequence, n will take thepositiveintegervalues1,2,3,... in that order.
1. (a) (5 marks) State the de.nition of a function F :Rn Rm is di.erentiable at a =(a1,. ..,an)Rn .
(b)
(5 marks) State the de.nition of a function F : Rn Rm is continuously di.erentiable at a = (a1,. ..,an)Rn .

(c)
(20 marks) De.ne the function F :R2 R by


F(x,y)=
.. .
xy if(x,y).
=(0,0)
.x2 +y2
.
0 if(x,y)=(0,0)
Determine with proof if F is di.erentiable at (0,0) or not. Determine with proof if F is di.erentiable at (1,0) or not.
2. (25 marks) Consider the system of equations y vx +wy
exy .sin(vw)= eand e we= .x.
Show that near(x,y,v,w)=(0,0,0,.1), (x,y)canbe expressed asa di.erentiable functionof(v,w) .x .y
and .nd thevaluesof and at(v,w)=(0,.1).
.v .w

0
k=0
1

2.0.5 sin(x
(.1)k 3. (20 marks) Prove that


x)dx =


x .
(2k+1)!(3k+2+ 2)
4. (a) (5 marks) Given an example of a sequence {an}of real numbers such that lim(an .2an+1 +an+2)=0 and lim(an .an+1).=0.
nn
(b) Let {an}be a bounded sequence of realnumbers and lim(an .2an+1 +an+2)=0.
n
(i) (15 marks) Prove that limsup (an .an+1) 0.
n
(ii) (5 marks) Using (i) or otherwise, prove that liminf (an .an+1) 0. Determine lim(an .an+1).
n n
Notations: R denotes the set of all real numbers. When n appears in the subscript of a sequence, n will take thepositiveintegervalues1,2,3,... in that order.
1. (25 marks) De.ne the function F :R2 R by
2
.xy2/(x2 +y.
2)if(x,y)=(0,0)
F(x,y)= .
0 if(x,y)=(0,0)
Determine with proof if F is di.erentiable at (0,0) or not. Determine with proof if F is di.erentiable at (1,0) or not.
2. (25 marks) Consider the system of equations
xw
xeyv .ye =1+ vw and v sin(xv)= wsin(yw).
Show that near(x,y,v,w)=(1,0,0,1), (x,y)canbe expressed asa di.erentiable functionof(v,w)and .x .y
.nd thevaluesof and at(v,w)=(0,1).
.v .w
3. (a) (5 marks) Let E be a nonempty set. State the de.nition of Sn : E R converges uniformly on E to S :E R. (Be sure to de.ne the sup-norm of a function.)

1
ex .1 .ex .12
(b) (20 marks) Determine lim . Prove that dx = . .
x0+ x xk!(2k+1)
0 k=1


4. (a) (5 marks) State the de.nition of the limit superior of a sequence {xn}of real numbers. (Be sure to state the de.nition of the Lset ifyou use the de.nition taught in class.)

n
(b) (20 marks) Let {an}bea sequenceofpositive realnumbers. Prove thatif limsup an =1, then
n

n
limsup a1 +a2 ++an =1.
n
Notations: R denotes the set of all real numbers. When n appears in the index or subscript of a sequence, n will take thevalues1,2,3,. ...
1. (a) (5 marks) State the de.nition of a function F :Rn Rm is di.erentiable at a =(a1,. ..,an)Rn .
(b) (20 marks) De.ne the function F :R2 R by
.xy/(x2 +y2)if(x,y)=(0.,0)

F(x,y)= .
0 if(x,y)=(0,0)
Determine wit