Lindsay Erickson
lae241 at
math.washington dot edu
office hours:
W 4-5 in PDL C-8-D
Th 1-2 in MSC
Midterm 1:
Tuesday, April 20
Midterm 2:
Thursday, May 13
Final Exam:
Saturday, June 5
1:30-4:20 PM
CDH 135
Course grades are now available on Catalyst. These are the grades I will be reporting to the registrar.
Final exam stats: n=33; min=39; 1st quartile=67; median=73; 3rd quartile=82; max=87 (out of 90)
UPDATE: My office hours this Friday will be 12-2 PM.
I will be holding an office hour on Friday from 12 to 2 PM if you would like to stop by and see your final exam. I will most likely have course grades calculated at that time.
Have a good break!
Here is a review sheet for the final exam.
Answers to some Taylor homework problems can be found here.
Writing problem #3:
Let f(x) be a function and I be an interval around 0.
Suppose f and all of its derivatives are defined in the interval I.
(a) Show that if f is even, then the Taylor series for f consists only of even terms, and if f is odd, then the Taylor series for f consists only of odd terms.
(b) Use Taylor series to (formally) show that
ei θ = cos θ +i sin θ
where i is the base of the complex numbers, so i2=-1.
Conclude the famous identity
ei π=-1.
Due June 4.
Here is an applet I wrote to illustrate Taylor polynomials of sin x.
We are now starting the Taylor Series portion of the course. Instead of using the textbook, our work will be based on these notes. I will endeavor to make the lectures as complete as possible, but these notes will be available to fill in gaps and give you more examples and other perspectives on the concepts.
Stats for the second midterm: n=33; min=27; 1st quartile=39; median=46; 3rd quartile=49; max=50 (7 students)
Here are some suggested problems for study from my exam archive:
Here is a review sheet for the upcoming midterm exam.
David asked a good question today: how do we know that the acceleration
vector is in the same plane as the unit tangent and unit normal vectors?
This hinges on the fact that the acceleration vector is a linear combination
of the unit tangent and unit normal vectors.
In general, if a vector a
is a linear combination of vectors b and c, then we can show that
a is in the same plane as b and c. How do we do this?
One way is to show that a is orthogonal to b x c. We do that
by letting
a = m b + n c.
Then we write
(m b + n c) dot (b x c) =
m b dot (b x c) + n c dot (b x c) = 0
this last due to the fact that a scalar multiple of a vector is orthogonal
to any cross product involving that vector.
Now, we need to show that the acceleration vector is a linear combination of
the unit tangent and the unit normal vectors.
We did this in class with a
fancy equation that explicitly showed that linear relationship.
More directly, we might do this.
We know
T(t) = r'(t)/|r'(t)|=r'(t)p(t), for a scalar function p(t), say,
so
T'(t) = r''(t)/|r'(t)| + r'(t) k(t)
for some scalar function k
(differentiating T(t) is kind of a pain, but this
is all we need to conclude about T'(t) for now).
So then N(t) = T'(t)/|T'(t)| = r''(t) m(t) = r'(t) n(t), say, where m(t) and n(t) are some scalar functions.
So, we've shown that T(t) = r'(t) p(t), and
N(t) = r''(t)m(t)+r'(t)n(t)
but r''(t) is the acceleration vector, so rearranging we can write
r''(t) as a linear combination of N and T,
and so the acceleration vector must be in the same
plane as N and T.
Whew!
The writing problem is now due on Monday, April 26.
All of Spring 2009 (in the archive at right) is worth examining in preparation for Tuesday's midterm.
In addition, the following problems on the old midterms available at right are worth studying:
Here is a short review sheet for midterm one.
Writing Problem #1: Give a thorough analysis of the
polar curves
rn = sin m θ
and
r = (sin mθ)n
where n and m are positive integers.
Some of these curves are referred to as "roses".
This applet displays some of these curves, plus others.
As a sort of summary of our beginning discussion of 3d geometry, here is a list of things you should be able to determine or find:
Welcome to Math 126C.
Announcements and other important information will appear here, so check back frequently.