Sections on the final, and some practice problems

The book sections covered on the final will be a subset of those covered in the homework, the last one of which was due Monday.

In particular, there is no homework nor will there be final exam questions about Laplace transforms nor series solutions to differential equations.

Here are some questions on most of the topics on the final to get you started.


1. Let A be a square matrix with A*A = Identity. Let M be a square matrix, such that AMA = M. Let v be an eigenvector of M with eigenvalue e.

a) Show that Av is one too.

b) Assuming that, show that v+Av, v-Av are too.


2. Find the exponential of t times the matrix M

[2 0 3 9 0]
[0 2 1 6 0]
[0 0 2 0 0]
[0 0 0 2 0]
[0 0 0 0 3]

Hint: it would be easier if you subtracted 2 * Identity... how do you make use of that?

Solve the ODE system y' = My for arbitrary initial conditions.


3. Let y' = f(t,y).

a) What is the general form of f, if this equation is separable and linear homogeneous?
b) What is the general form of f, if this equation is separable and linear but not necessarily homogeneous?


4. Let c_0 = 0, c_{n+1} = c_n + 2n + 1. Find c_0,c_1,c_2,c_3,c_4,c_5 and guess a general formula for c_n. Then determine the radius of convergence of the power series \sum_n c_n x^n.



5. Assume that
f(x) y dx + (x+y) dy = 0
is exact. Determine the possibilities for f, and solve this ODE. (You should have two constants in your answer: one because you don't know f precisely, and the other from solving the ODE.)

HW #7 due Monday Nov 28th (revised)

4.2 #12 11,13-14,20 ("characteristic" = "eigen")
4.3 #2-3
4.5 #1,2,7

And now for something completely different

If you're curious what I do in the other half of my professor gig, on Thursday I'm giving an undergrad talk on these:




Puzzles, straightening honeycombs, and adding matrices

Given the three kinds of puzzle pieces pictured above, define a puzzle to be a decomposition of a triangle into puzzle pieces (such that the edges match up, like in a jigsaw puzzle). Call a puzzle ``rigid'' if there is no other puzzle with the same outer boundary. A lot can be proven about puzzles (we'll do (1) and (2) in the talk):

1. The number of 0s on one side equals the number of 0s on each of the other two sides - see if you can prove this one before the talk!

2. The lines in the puzzle pieces can all be simultaneously straightened (as in the right-hand picture) if and only if the puzzle is rigid

3. There is an easy 1:1 correspondence between rigid puzzles and inequalities on the eigenvalues of the sum of two Hermitian matrices

4. The statement ``Given four generic lines in space, there are exactly two others that touch all four,'' and others like it, can be turned into puzzle-counting statements.


Thursday, November 17, 12:00 pm, AP&M 2402
Refreshments provided!

HW #6

3.1 #1-3,7-9,10-14
4.1 #1-4

Dropping one homework

This is something I meant to declare a few weeks ago; silly of me.
We're going to drop one homework for each person when calculating grades.

Tests are graded

People definitely did better on this test than the last one. Still unpleasantly many uses of the "universal distributive law", e.g. "e to the a+b equals e to the a
plus e to the b", or "(x^2 + y^2)^n = x^2n + y^2n". Algebra misteakakes are a real killer in this (and basically any other) math class.

The grade boundaries (remember these are approximate, and that the eventual grade is calculated from the number, not the letter):
00-10 F
10-20 D
20-35 C
35-50 B
50-up A
Way more As and fewer Fs this time.

I'll have the answers posted later in the evening, and sometime soon the grading scheme for each problem. The tests will be back tomorrow.

Book questions to prepare for the midterm

Conrad
1.1 #14-21
1.2 #5-7, 8-15 (not the graphing part so much)
1.3 #13-16, 26-30 (again, skip the graphing)
2.1 (skip the friction stuff, which we didn't discuss) #8-15
2.2 #1-14, 22
2.4 #1-7, 12-21

Truly, I am sorry about the delay in getting this stuff on the web. I do have a doctor's note.

Midterm #2 Wednesday

Sorry for the delay in posting this stuff about midterm #2; some family stuff has occupied me continuously since 2 PM yesterday. I will not be able to post a list of sample problems until morning afternoon.

There will be no Matlab this Tuesday; rather, the TAs will be in their offices for additional office hours during that time. Nor shall there be HW due Friday (Veteran's Day); instead, the homework due the following Friday will be slightly longer.

The sections covered will be all in Conrad (no sequences/series):
1.1, 1.2, 1.3, 2.1, 2.2, 2.4. We've also done a little graphical analysis of ODEs in class that might show up (or might not).

Same setup: no books (including blue), no calculator, one double-sided handwritten sheet that you shouldn't bother filling with integral tables. And as on the last test, learn the rules of exponents or you may hurt your chances very badly.