Math 31BH Homework 6 Solutions
March 3, 2015
3.2.1
(a) The smooth curve can be seen as the kernel of the function
x
y
F
and therefore its tangent line at
1
0
= x + x2 + y 2 2,
has equation
x1
= 0.
y0
1
0
DF
Now
DF
x
y
= 1 + 2x 2y
so
DF
1
0
= 3 0
and there
Problem 1.
Consider the function
f (x, y) = 3y 2 2y 3 3x2 + 6xy.
Find the critical points of the function and determine their nature.
We calculate
fx = 6x + 6y = 0 = x = y
fy = 6y 6y 2 + 6x = 0 = x + y = y 2 = 2y = y 2 = y = 0 or y = 2.
We nd the second d
Math 31BH - Winter 2014 - Midterm II Solutions
Problem 1.
Find the critical points of the function
f (x, y) =
2 3
x + x2 y y 2 3y
3
and determine their nature.
Solution:We calculate the derivatives of f and set them to 0:
fx = 2x2 + 2xy = 0 = x(x + y) = 0
MATH 31BH Homework 6 Solutions
February 21, 2014
Problem 3
(i) The Taylor expansion of the exponential function around a = 0 is
ex = 1 + x +
x2 x3
xn
+
+ . +
+ .
2!
3!
n!
Proof. Let f (x) = ex . We need only nd f (n) (0) for each n N cfw_0.
We claim that
Math 31BH, Homework Assignment #3, due February 3, 2015
Do the following problems from the textbook:
1.5.21cde (you may want to recall that lim x ln x = 0 , which can be proved with
x0
LHopitals Rule),
0.5.2, 1.7.2, 1.7.4ab (in part b, the given definitio
Math 31BH, Homework Assignment #6, due March 3, 2015
(Note that you have TWO weeks to do this assignment!)
Do the following problems from the textbook:
Section 3.2, exercises 1, 3, 4, 6, 7.
Section 3.3, exercises 1, 6, 13.
(For exercise 3.3.6, the Taylor
Math 31BH, Homework Assignment #1, due January 20, 2015
1. Prove that a sequence of real numbers cannot converge to two distinct limits. That
is, if a b , then a and b cannot each be a limit of the sequence.
2. Show that there exists a rational number bet
Math 31BH, Homework Assignment #2, due January 27, 2015
From the textbook, do Exercises 1.5.1, 1.5.4, 1.5.5, 1.5.7bcd, 1.5.13, and 1.6.1. Problem
1.5.4a, the interior of A is the biggest open set contained in A, should be interpreted as
meaning: any open
Math 31BH, Homework Assignment #4, due February 10, 2015
Do the following problems from the textbook:
Section 1.8, problems 2, 7, 8, 11
(For problem 8: if such a function g existed, what would the Chain Rule imply?)
Chapter 1 review exercises (p. 155), pr
MATH 31BH HOMEWORK 8 SOLUTIONS
Problem 3.1.2
The set M =
x
y
R2 : x + x2 + y 2 = 2
is a smooth curve.
Proof. Consider the function F : R2 R dened by
F
x
y
= x + x2 + y 2 2.
Now M is exactly the set of points for which F
x
y
= 0.
Notice that
[DF ] = [1 +
MATH 31BH HOMEWORK 7 SOLUTIONS
Problem 3.7.4 (b)
Let X Mat(2, 2) be the set of 2 2 matrices with determinant equal to 1. In other words, let
X=
x y
: xw yz = 1 .
z w
We seek a matrix in X that is as close as possible to
x y
0 1
z w
1 0
0 1
. Therefore we
SOLUTIONS TO PROBLEMS WHICH ARE NOT IN THE SOLUTIONS MANUAL.
4.
(i) We have
fx = 2xy 4 + y 2 ln(2x y) + xy 2
2
= fx (1, 1) = 4.
2x y
Next,
1
= fy (1, 1) = 3.
2x y
Thus f (1, 1) = (4, 3). The direction of steepest increase is (4, 3).
(ii) We have
1
Dv f (
Math 31BH Homework 5 Solutions
February 17, 2015
3.1.2
x
R2 : x + x2 + y 2 = 2 is a smooth curve.
y
Consider the function F : R2 R dened by
The set M =
F
x
y
= x + x2 + y 2 2.
Now M is exactly the set of points for which F
x
y
= 0.
Notice that
[DF ] = [1
Math 31BH, Homework Assignment #7, due March 10, 2015
1. Consider the function f (x, y) = 3xe y x 3 e3y on R2.
(a) Show that this function has exactly one critical point, which is a local
maximum.
(b) Show, however, that this point is not an absolute maxi
Math 31BH Prof. Rabin Winter, 2015 Course Information
Course: Math 31BH, Honors Multivariable Calculus, MWF 10:00 10:50 AM in APM B412. You
should be enrolled in the discussion section, which meets on Tuesdays at 2 PM in APM B402A. The
first section meeti
Calculus II
Preface
Here are my online notes for my Calculus II course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus II or needing a refresher in some o
Math 31BH - Winter 2014 - Final Exam
Problem 1.
Consider the function f (x, y, z) = 1 + x2 y 2 exp(x + y + 2z).
(i) At the point P (1, 1, 1), nd the unit direction of steepest increase for f .
(ii) For the vector u = i + 2j + k, nd the directional derivat
MATH 31BH Homework 5 Solutions
February 14, 2014
Problem 1.8.2
(a) Let
x
t
f y = x2 + y 2 + 2z 2 and g(t) = t2 .
t3
z
Then by the chain rule
a
a
a
D(g f ) b = Dg f b Df b
c
c
c
= [Dg(a2 + b2 + 2c2 )]
1
= 2a2 + 2b2 + 4c2
3(a2 + b2 + 2c2 )2
2a
=
Problem 1.
Find the critical points of the function
f (x, y) = 2x3 3x2 y 12x2 3y 2
and determine their type i.e. local min/local max/saddle point. Are there any global min/max?
Solution: Partial derivatives
fx = 6x2 6xy 24x, fy = 3x2 6y.
To nd the critica
I wanted to make a few remarks about topics that have come up recently, without taking
up class time to do so.
First, I am trying to balance the theoretical aspects of the course with the computational
ones. By then end of the course I want you to be able
MATH 31BH Homework 3 Solutions
January 21, 2014
Problem 1
Let f : R R be a dierentiable function such that f is bounded. Then there exists a constant
M > 0 such that for all real numbers x and y we have
|f (x) f (y )| M |x y |.
Proof. Since f is bounded,
MATH 31BH Homework 1 Solutions
January 10, 2014
Problem 1.5.2
(a) (x, y )-plane in R3 is closed and not open.
To see that this plane is not open, notice that any ball around the origin (0, 0, 0) will contain
points whose third component is not 0, and henc
Math 31BH - Homework 3. Due January 29.
Part I.
1. (Wednesday.) A function f : R R is said to be Lipschitz if there exists a constant M > 0
such that for all real numbers x, y we have
|f (x) f (y )| M |x y |.
Use the mean value theorem to show that if f i
Math 31BH - Winter 2011 - Midterm I
Name:
Student ID:
Instructions:
Please print your name and student ID.
During the test, you may not use books or notes.
Read each question carefully, and show all your work. Answers with no explanation will receive no c
MATH 31BH Homework 4 Solutions
February 7, 2014
Problem 1
If T : Rn Rm is a linear map given by T (x) = A x for some m n matrix A, then T admits
a total derivative which is equal to the matrix A.
Proof. We show that A ts the denition of the total derivati
Problem 1.
Find the limits below or explain why they do not exist:
(i) limx,y0
(x2 +y 2 )2
2x2 +3y 2
We note that
First, the function
x2 + y 2
(x2 + y 2 )2
=2
(x2 + y 2 ).
2x2 + 3y 2
2x + 3y 2
x2 +y 2
2x2 +3y 2
is bounded by 1 . Indeed,
2
x2 + y 2
1
2(x
Math 31BH - Winter 2014 - Midterm I
Problem 1.
Show that for all x 0, the following inequality holds
x ln(1 + x) 0.
Solution:Consider the function
f (x) = x ln(1 + x).
We have
x
1
=
0
1+x
x+1
for x 0. Thus, f is increasing over [0, ) hence for x 0 we have
Math 31BH - Winter 2014 - Midterm I
Name:
Student ID:
Instructions:
Please print your name and student ID.
During the test, you may not use books or notes.
Read each question carefully, and show all your work. Answers with no explanation will receive no c