MATH 222, Calculus 3, Fall 2011
Solutions to Assignment 1
1. For each of the following sequences (an ), state whether it converges or
diverges. For each one that converges, nd the limit.
(a)
an = cos(
1
+ 2 ).
6
n
Solution: Since the cosine is continuous,
COMS 210Fall 2009
Introduction to Communication Studies
Lecture: MW 11:35-12:25 + 1 weekly conference (beginning 16 September)
(You are required to register for your conference. Check Minerva for available days and times.)
(Both the lecture sections of th
COMS210
Introduction to Communication Studies
Fall 2010
As the only required course in our minor, COMS 210 offers an introduction to the field of
Communication Studies as it is practiced at McGill. Students wil
AEMA 102, Calculus II
FINAL EXAMINATION
Date: April 27, 2010
Examiners: Time: 14:00 - 17:00
Profs. J.F. Ha es and RJ. Cue
Use of calculators is prohibited.
-MW
1. Determine if each of the following sequences {an} converges or diverge
1
Eigenvectors continued
The eigenvector train continues on and stops for no one.
OK, as most of you gured out, the eigenvalues of a matrix are the solutions
of the equation
det(A I ) = 0
and the eigenspace Ei is the space N (A i I ).
Let A be an nn matri
1
Eigenvalues and eigenvectors - wtf ?
In general, a linear transformation T maps any line LRn to some other line in
Rn . Of course, there are a bazillion dierent lines in Rn , so there is no systematic way to understand T by looking at random lines. The
MATH 222, Calculus 3, Fall 2011
Solutions to Assignment 3
1
1. Find the Maclaurin series rst for (1 t2 ) 2 and then for arcsin x =
sin1 x. (Hint: integrate.) Use this give another formula for .
4
Solution: From the binomial series (1 + x)m =
n=0
m
n
xn ,
MATH 222 HOMEWORK 2
DUE SEPTEMBER 26, 2011
1. P ROBLEMS
Problem 1.1. Find the interval of convergence.
(3n + 2)xn
(a.)
n=1
(8x)n
n!
(b.)
n=0
(n + 2)2 xn
(c.)
n=0
100
(d.)
n=1
xn
n
Solutions
Part a.) Set an = (3n + 2)xn . The series has the form an . The r
MATH 222 HOMEWORK 4
DUE OCTOBER 11, 2011
October 10 is a holiday. The assignment is due by 5PM on October 11.
Rogers Section - Drop off the assignment outside the door of 1243 Burnside Hall.
Loveys Section - Drop off the assignment outside the door of 916
MATH 222, Calculus 3, Fall 2011
Assignment 3, due in class Monday, October 3, 2011
1
1. Find the Maclaurin series rst for (1 t2 ) 2 and then for arcsin x =
sin1 x. (Hint: integrate.) Use this give another formula for .
4
2. Find a unit vector perpendicula
MATH 222 HOMEWORK 2
DUE SEPTEMBER 26, 2011
1. P ROBLEMS
Problem 1.1. Find the interval of convergence.
(3n + 2)xn
(a.)
n=1
(8x)n
n!
(b.)
n=0
(n + 2)2 xn
(c.)
n=0
100
(d.)
n=1
xn
n
Problem 1.2. Find the Maclaurin series for f (x).
(a.)
f (x) = sin (x) + x
MATH 222, Calculus 3, Fall 2011
Assignment 1, due in class Monday, September 19, 2011
1. For each of the following sequences (an ), state whether it converges or
diverges. For each one that converges, nd the limit.
(a)
an = cos(
(b)
an =
n2
1
+ 2 ).
6
n
Compute double integrals in polar coordinates
Useful facts: Suppose that f (x, y ) is continuous on a region R in the plane z = 0.
(1) If the region R is bounded by and a r b, then
beta
b
f (x, y )dA =
f (r cos , r sin )rdrd.
R
a
(2) If the region R is bo
MATH 222, Calculus 3, Fall 2011
Assignment 7, due in class Monday, November 7, 2011
1. Your box has to be able to hold 40m3 of stu. It does not need a top.
Material for the sides is relatively cheap at $1 per square meter; the bottom
material is more expe
MATH 222 HOMEWORK 6
DUE OCTOBER 31, 2011
1. P ROBLEMS
Problem 1.1. There is only one point where the plane tangent to the surface z = x2 + 2xy +
2y 2 6x + 8y is horizontal. Find it.
Problem 1.2. Find an equation of the plane tangent to the surface z = f (
MATH 222, Fall 2011
Calculus 3
Assignment 3, due Friday, October 21, 2011
1. For the helix dened parametrically by
r(t) = 3 cos t, 3 sin t, 4t ,
(a) nd the arc length s(t) travelled from 0 to t > 0, and specically
from 0 to 6 ;
(b) nd the parametrization
MATH 222 HOMEWORK 8
DUE NOVEMBER 14, 2011
1. P ROBLEMS
Problem 1.1. Find the area of the region bounded by the curves y = x2 + 1 and y = 9 x2 .
Problem 1.2. Find the volume of the solid bounded by the surfaces z = x2 + 2y 2 and
z = 12 2x2 y 2 .
Problem 1.
PHYS257 REPORT WRITING GUIDELINES
SCIENTIFIC REPORTS
(A)
OVERVIEW
Experimental data become useful to others only when these results are communicated
effectively. The two primary ways in which new results are communicated to the scientific
community at lar
Physics 257: Assignment 0
Read Chs. 1 and 2 of the course text (I. Hughes and T. Hase, Measurements and their uncertainties
(Oxford, 2010) before starting this assignment. These two chapters will introduce you to mean,
standard deviation and significant f
PHYSICS 230: Dynamics of Simple Systems
Assignment #1
Due Thursday Sept 22 by 16:00
You may hand the assignment to me before or after class but not during the lecture,
or you may hand the assignment in at my ofce. Late assignments will not be accepted,
wi
PHYSICS 230: Dynamics of Simple Systems
Assignment #2
Due Thursday Oct 6 by 16:00
You may hand the assignment to me before or after class but not during the lecture,
or you may hand the assignment in at my ofce. Late assignments will not be accepted,
with
Vectors Everywhere
more than just the usual, familiar magnitude and
Vectors
direction business
direction business
well-known applications: e.g. wind velocity
Geometry & Algebra of Vectors;
Scalar and Cross Products
Geometric Vectors
Vectors are represe
Linear systems
Linear equation: 3x + 5y - 18z = 0
Systems of Linear Equations
(Contrast this to 3x2 + 5yz = 0, which is NOT linear)
Sets of equations:
Introduction & Methods
Solution sets : case 1
Infinite number of solutions (consistent), with some
fr