Math247
Assignment 1 Solutions
Winter 2012
1. Define, for x Rn ,
kxk = max |xi |.
1in
(a) Prove that k k defines a norm.
(b) Show that for all x Rn ,
1
kxk2 kxk kxk2 .
n
Also show that equality holds
Math 247, Spring Term 2012
Homework Assignment 7 Solutions
Problem 1. Let A be the bounded subset of R2 dened as follows:
A :2 cfw_(s,t) 6 (0,1] X (0,1] ls S t.
Consider the function f : A % R dened b
Math 247, Fall Term 2012
Homework Assignment 3 Solutions
For Problem 1, recall that a function f : R" > Rm is said to be linear when it respects
vector space operations:
f(0153+)= af(fl+f(37), VCYHB 6
Math 247 , Spring Term 2012
Homework Assignment 2 Solutions
Problem 1. Let (:Ek >211 and 55 be vectors in R", and considerthe following condition
which they may satisfy:
(EV-CT) 31% E N such that \75
Math 247, Spring Term 2012
Homework assignment 9 Solutions
Problem 1. Consider the function f : R2 + IR dened by
_ s2t/(32+t2) if (s,t)7$(0,0)
f(s,t)-cfw_ 0 if (s,t)=(0,0).
This is the same function a
Math 247, Spring Term 2012
Homework Assignment 1 Solutions
In Problem 1, let :3 and 37 be two non-zero vectors in R". We consider the Cauchy-
Schwarz inequality that was proved in class (Proposition 1
Math 247, Spring Term 2012
Homework Assignment 5 Solutions
Problems 1-3 are about the oscillation of a function on a set, Recall that if f : A A R
is a bounded function and if B is a nonempty subset o
Math 247, Spring Term 2012
Homework Assignment 6 ~ Solutions
Problem 1. Let A1, . . . ,AT be nonempty bounded subsets of R such that A,- Aj = (Z)
for z' 75 j. Suppose that for every 1 S 2' g 7 we are
Math 247, Fall Term 2011
Homework Assignment 8 Solutions
Problem 1. Let b be a positive real number, and let D be the punctured disk of radius
17 in R2:
D = cfw_(s,t) e R2 10 < \/82+t2 g b.
By using p
Math 247, Spring Term 2012
Homework Assignment 10 Solutions
Problem 1. Let f : R > R be a 01 function (that is, f is differentiable and f : R > R
is continuous). Let F : R2 > R be the function dened b
Math 247, Spring Term 2012
Homework assignment 4 Solutions
Problem 1. Consider the following equivalence which was stated in class (in Lecture
4, Proposition 4.4): given A Q R", a function f : A -) Rm
Math 247, Fall Term 2012
Homework Assignment 1 Solutions
In Problem 1 we consider the CauchySchwarz inequality that was proved in class:
(C-S) 1(f,37)lSllfll-llll-
Problem 1. Let f and g be two vector
Math 247
Assignment 4
Due Friday, October 24, 2014
1. Let
x5
if (x, y) 6= (0, 0)
f (x, y) =
(y x2 )2 + x8
0
if (x, y) = (0, 0).
(a) Show that Du f (0, 0) exists for every unit vector u.
(b) Show that
Math 247
Assignment 2
Due Friday, October 3, 2014
n
1. Suppose the sequence cfw_xk
k=1 R satisfies |xk+2 xk+1 | |xk+1 xk | for all
k 1, with 0 < < 1. Show that cfw_xk is a Cauchy sequence and hence
Math 247
Assignment 3
Due Friday, October 10, 2014
1. Let S Rn be both open and closed. Show that S = Rn or S = .
2. For a Rn , r > 0, n 2, show that Er (a) = cfw_x Rn : |xa| > r is path-connected
and
Math247
Assignment 4 Solutions
Winter 2012
1. Consider f : Rn Rn such that there is 0 k < 1 with
kf (x) f (y)k kkx yk
for all x, y Rn . (Such a function is said to be a contraction. ) For arbitrary x0
Math247
Assignment 2 Solutions
Winter 2012
P
n
1. Let cfw_yi
i=1 R and suppose that
i=1 kyi k < . For j 1 define the sequence of
vectors
sj = y 1 + y 2 + . . . + y j .
n
Prove that cfw_sj
j=1 is a c
Math247
Assignment 3 Solutions
Winter 2012
1. Let
(
f (x, y) =
x5
(yx2 )2 +x8
if (x, y) 6= (0, 0)
if (x, y) = (0, 0).
0
(a) Show that Du f (0, 0) exists for every unit vector u.
(b) Show that f is not
Math247
Calculus 3 (Advanced)
Winter 2012
Assignment #3
Due:
1. Let
(
f (x, y) =
x5
(yx2 )2 +x8
10am January 31, 2012
Drop Box #3 Slot #8
if (x, y) 6= (0, 0)
if (x, y) = (0, 0).
0
(a) Show that Du f (
Math247
Calculus 3 (Advanced)
Winter 2012
Assignment #1
Due:
10am January 13, 2012
Drop Box #3 Slot #8
1. Define, for x Rn ,
kxk = max |xi |.
1in
(a) Prove that k k defines a norm.
(b) Show that for a
Math247
Calculus 3 (Advanced)
Winter 2012
Assignment #2
Due:
10am January 20, 2012
Drop Box #3 Slot #8
P
n
1. Let cfw_yi
i=1 R and suppose that
i=1 kyi k < . For j 1 define the sequence of
vectors
sj
Math247
Calculus 3 (Advanced)
Winter 2012
Assignment #4
Due:
10am February 7, 2012
Drop Box #3 Slot #8
1. Consider f : Rn Rn such that there is 0 k < 1 with
kf (x) f (y)k kkx yk
for all x, y Rn . (Suc
Math 247
Assignment 7
due Monday, November 17, 2014
1. Suppose that f : O Rn Rn , O open and convex, f C 1 (O), and the symmetric
(Df )(x) + (Df )T (x)
part of the Jacobian matrix,
, is positive defin
Math 247
Assignment 5
Due Friday, October 31, 2014
1. Let f : Rn \ cfw_0 R be homogeneous of degree p: f (tx) = tp f (x) for all x Rn \ cfw_0
and for all t > 0, where p is a constant.
(a) For f C 1 (R
Math 247
Assignment 6
Due Friday, November 7, 2014
1. Let f : O Rn R, f C 2 (O), O an open convex set. Assume that D2 f (x) is
positive semi-definite x O. Such f are said to be convex functions.
(a) P
Math 247, Spring Term 2012
Homework Assignment 10
Posted on Thursday, July 19; due on Wednesday, July 25
Problem 1. Let f : R R be a C 1 function (that is, f is differentiable and f 0 : R R
is continu
1
Math 247, Spring Term 2012
Homework Assignment 1
Posted on Wednesday, May 2; due on Wednesday, May 9
In Problem 1, let ~x and ~y be two non-zero vectors in Rn . We consider the CauchySchwarz inequal
Math 247, Spring Term 2012
Homework assignment 7
Posted on Wednesday, June 27; due on Wednesday, July 4
Problem 1. Let A be the bounded subset of R2 defined as follows:
A := cfw_ (s, t) (0, 1] (0, 1]
Find the derivative of y = sin"1(3:). '
SM? :1 3\V\ SN;l :1 X
0 A i ( i
(Six SM? 3 0W x
COS/:1) 3" :
\ ._
E9 (ogj' Find the derivative of y : tanKw). 1
Find the derivative of y = sin (632 .