HOMEWORK #9
DUE FRIDAY APRIL 16TH
(1) Determine whether the following series converges and whether it converges absolutely.
n
(a) (1!i)
n=1
n
(b) (1/3 + 2/3i)n
n=1
(2) Use the ratio test to nd the radius of convergence of the following power series.
Justi
HOMEWORK #4
DUE WEDNESDAY FEBRUARY 3RD
(1) Give example of the following types of sequences.
(a) An unbounded sequence that has a convergent subsequence.
(b) A bounded sequence that has two dierent convergent subsequences which converge to
distinct values
HOMEWORK #6
DUE MONDAY MARCH 8TH
(1) Write down matrix representations of the following linear transformations. Also
explain as well as you can what this linear transformation does geometrically. Fix an
orthonormal basis u, v for R2 and an orthonormal bas
HOMEWORK #7
DUE FRIDAY MARCH 19TH
(1) Fix u, v R2 to be a basis. Find the eigvalues and describe the eigenvectors of the
following linear transformations.
(a) The map T : R2 R2 dened in the following way. T (u) = v and T (v) = 2v.
10
(b) The map T : R2 R2
HOMEWORK #8
DUE FRIDAY MARCH 26TH
(1) Are the following matrices diagonalizable? If so, nd a diagonal matrix similar to
them. If not, justify why they arent diagonalizable.
13
(a)
02
13
(b)
01
12
(c)
(Hint, look at the solutions to an old worksheet)
34
(2
HOMEWORK #3
MATH 1861
WINTER 2010
DUE WEDNESDAY JANUARY 27TH
(1) Suppose that f exists.
(a) Show that
f (a + h) + f (a h) 2f (a)
h0
h2
Hint: Use the Taylor polynomial P2,a (x) with x = a + h and with x = a h.
(b) Let
x2 ,
x0
f (x) =
x2 , x < 0
f (a) = lim
SYLLABUS
Math 186 Section 1 Winter 2010
Instructor: Karl Schwede
Class web page: http:/www-personal.umich.edu/kschwede/math186
Text: Calculus by Michael Spivak
Suggested Text: Calculus and Linear Algebra by Wilfred Kaplan and Donald Lewis
Contacting
PRACTICE FOR EXAM #1
1. Write down the Taylor series for the following functions centered at a.
2
(a) f (x) = e(x ) centered at a = 0.
f ( x) =
n=0
(x2 )n
=
n!
n=0
x2n
n!
Note the uniqueness of power series (see the last page of problems) implies that thi
NOTES ON INJECTIVE AND SURJECTIVE FUNCTIONS
MATH 1861
WINTER 2010
First we recall the denition of a function.
Denition 0.1. A function is the following information.
(a) A domain = D. In other words, a set of allowable input values.
(b) A codomain = C . In
WORKSHEET ON EIGENVALUES AND EIGENVECTORS
MATH 1861
Denition 0.1. Suppose that T : Rn Rn is a linear transformation.1 A non-zero vector v Rn
is called a eigenvector for T if there exists a number such that T (v) = v. In this case, the
number is called an
INTRODUCTION TO DIFFERENTIAL EQUATIONS
MATH 1861
1. Ordinary differential equations
We work with real numbers in this worksheet.
Denition 1.1. Fix x to be a variable, and y : [a, b] R to be an unknown function (of x).
An ordinary dierential equation is an
WORKSHEET ON SIMILAR MATRICES, EIGENVECTORS AND
CHARACTERISTIC POLYNOMIALS
MATH 1861
Denition 0.1. Two 22 matrices A and B are called similar if there exists a linear transformation
T : R2 R2 such that both A and B represent T but with respect to dierent