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
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 +
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
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 ne
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)
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.
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 se
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 cont
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 Cha
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 o
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 c
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 directio
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 se
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)
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
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 wantin
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) Fo
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 + b
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 deriv
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 computationa
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 ha
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
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
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 y
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.
Proo
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
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.
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 y