MATH 251
Maple Lab 2
FALL 2016
A webpage will be posted listing individual data for each student. The data for this lab
will be a vector-valued function. The components of the vector function will be various
combinations of sine and cosine, perhaps raised
Math 251
Maple assignment #4
FALL 2016
Code and Data
It's very important to distinct between data and code. Write your code so that the data and code
are separated.
If you change data, you do not need to change the code!
A website will be posted listing i
Math 251
Maple assignment #3
FALL 2016
You are encouraged to discuss this assignment with other students and with the instructors,
but the work you hand in should be your own.
A webpage will be posted listing individual data for each student. For this lab
Math 251
Maple assignment #5
FALL 2016
Code and Data
It's very important to distinct between data and code. Write your code so that the data and code
are separated.
If you change data, you do not need to change the code!
A website will be posted listing i
Workshop 5 Math 251, M. Naumova
Problem
Let x(t) be the curve given by
x(t) = (2t + t2 , 2t , t + t2 + t3 /3 ) .
Compute T(t), N(t), B(t) and the speed, curvature and torsion v(t), (t) and (t).
Is x(t) a planar curve? Justify.
Compute the function s(t) gi
Workshop 4 Math 251, M. Naumova
Problem
Prove that the length of a curve as computed using the arc length integral does not
depend on its parametrization.
Workshop 1 Math 251, M.Naumova
Problem 1
A plane is flying due east at 200 km/h encounters a 40 km/h wind blowing in the northeast direction.
The resultant velocity of the plane is the vector sum v = v1 + v2, where v1 is the velocity vector of the
plane a
Workshop 6 Math 251, M. Naumova
Problem
1
, find
For f1 (x, y) = (x2 + y 2 ) sin xy
For f2 (x, y) =
sin(xy)
,
x
For f3 (x, y) = 1 +
For f4 (x, y) =
find
lim
x0,y0
find
y x
,
x
2xy
,
x2 +y 2
0,
lim
x0,y2
find
lim
x0,y0
f1 (x, y).
f2 (x, y).
lim
x+,yk
f
Math 251, Solutions to Quiz 2
1. (5 points) Find length of the following curve over the given interval
r(t) = ln(t2 ),
1 2
t , 2t ,
2
1
t
2
For the velocity vector we have
r (t) =
r (t) =
2
, t, 2
t
4
+ t2 + 4
t2
Now, using the length formula
2
L=
2
2
+t
Math 251, Solutions to Quiz 1
Directions: Present a complete solution to the problems showing your work. No partial credit will be given
for wrong answers without intermediate steps.
1. (4 points) Find the decomposition of a with respect to b
a = a + a ,
Math 251, Spring 2015, Midterm I Study Guide
Midterm I will take place in lecture on Fri, Feb 27, and will cover Sec 12.1-12.5, 13.1-13.4, and 14.1-14.8 of the textbook.
Below is a list of types of problems you are expected to be familiar with. See the li
MATH 251
Maple Lab 1
FALL 2016
You are encouraged to discuss this assignment with other students and with the instructors,
but the work you hand in should be your own.
For your personalized data and helpful background material see the web page
http:/www.m
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
September, 2011
2
Contents
1 GEOMETRY, ALGEBRA AND THE ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES
1.1
1
n
Algebra and geometry in
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
September, 2011
2
Contents
1 GEOMETRY, ALGEBRA AND THE ANALYSIS OF FUNCTIONS OF SEVERAL VARIABLES
1.1
1
n
Algebra and geometry in
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
September, 2012
2
Contents
1 GEOMETRY, ALGEBRA AND Analysis IN SEVERAL VARIABLES
1.1
n
1
Algebra and geometry in R . . . . . . . .
Solutions for the rst 9 exercises from Chapter 4
Eric A. Carlen1
Rutgers University
October 15, 2012
4.1 Let v1 = (1, 1) and v2 = (0, 1). Let f : R2 R be dierentiable, and suppose that for some
x0 R2 , v1 f (x0 ) = 2 and v2 f (x0 ) = 2. For v = (2, 3), co
Math 251, Solutions to Quiz 3
1. (5 points) Evaluate the limit or determine that it doesnt exist
lim
(x,y)(0,0)
x2 + y 2
cos(
x2 + y 2 ) 1
Using the polar coordinates
x = r cos ,
y = r sin
we can rewrite the limit
lim
x2 + y 2
= lim+
r2 cos2 + r2 sin2
=
Math 251, Solutions to Quiz 4
1. (5 points)
Evaluate the double integral
R = [0, ] [0, ]
(sin x + cos y) dA,
R
First, we present the double integral as an iterated integral
(sin x + cos y) dx dy
(sin x + cos y) dA =
R
0
0
Taking the integral inside we get
Math 251, Quiz 5
Directions: Present a complete solution to the problems showing your work. No partial credit will be given
for wrong answers without intermediate steps.
1. (5 points)
Let P = (0, 0), Q = (2, 3), and R = (5, 1). Present the triangle P QR a
Math 251, Quiz 6
Directions: Present a complete solution to the problems showing your work. No partial credit will be given
for wrong answers without intermediate steps.
1. (5 points)
Find the integral using polar coordinates (, , )
z, dV
where the is the
Math 251, Spring 2015, Midterm II Study Guide
Midterm I will take place in lecture on Fri, April 10, and will cover Sec 12.7, 15.1-15.4, 15.6, and 16.1-16.3 of the textbook.
Below is a list of types of problems you are expected to be familiar with. See th