Overview of the Course
This is the third term of the Calculus 17 series. The first two
terms introduced the notions of limits, derivatives, and integrals, and
provided much practice in their application; you are expected to know
this material well. This term, we will scratch the surface of
three very interesting topics:
- Calculus with Multiple Variables: Here we will learn to apply
many of the techniques learned for functions of a single variable to
functions with many variables.
- Systems of Ordinary Differential Equations: Systems of ODE's are
of fundamental importance in many areas of application; we will learn
basic techniques of solution and modeling.
- Permutations, Combinatorics, and Probability: Probabilistic
problems occur in many places in biology, such as the interpretation of
noisy DNA data. A good foundation in combinatorics is essential
to tackling such problems.
This is quite a bit of material, and the focus of the class will
shift considerably when we move from one topic to the next. Remember
that the final will be cumulative, so forgetting material once we've
moved on is not an option!
Structure
The class meets MWF from 12:10-1pm in Olson 147, with a problem
session Thursday 5:10-6pm in Haring 2016. We will have office hours
twice a week, at a time to be determined during the first class; my
office is
MSB2145, on the second floor of the Mathematical Sciences
Building. Depending on the will of the class, we may occasionally
hold office hours in the spacious Yurt
at Baggins End. I will also endeavour to arrange review sessions
before the midterm and final. Please come and ask
questions!
Homework will be assigned but not collected. Instead, we will have
quizzes on alternating Fridays at the beginning of class. The quiz will
consist of problems assigned as homework; keep up with the problem sets
and good quiz grades will follow. The quizzes will be worth 30% of the
total grade for the class. There will also be one midterm (30%) and a
final (30%). There will be no makeup exams or quizzes. Finally, once
during the term each student will submit a thorough write-up of a more
difficult problem, chosen from a list on the course website. Full
details of this assignment will be posted on the website as well. This
problem write-up will count as 10% of the final grade.
Homework
Notes:
30 March: I wrote section '10.3' on the board for the
homework, but the assignment should be for section 10.2, as it says
below.
8 April: Here's some notes for the chain rule and directional derivatives.
The material is analagous to what's in section 10.5, but with the
extra-matrixy spin that we've been talking about in class. Really, I
just wanted to correct my board mistakes!
15 April: Challenge
problems! Currently there are five problems, and there are a couple
more I'll add soon. Notice there's an example write-up, and please,
please, please ask questions!
18 April: Notes on multivariable integration, from Paul's
Online Notes.
Double
Integrals
Iterated
Integrals
Double
Integrals over General Regions
We'll also use Kouba's worksheet
for double integrals as a source of homework
problems.
19 April:
Here's a practice midterm! It's
rather longer than the actual midterm will be, but you should be sure
that you can do anything on this.
26 April:
Here's some practice midterm solutions! Problems 8 and 9 were a bit on the impossible side. Try problem 8 with f=x^2+y^2, and problem 9 with f(x,y)=1, and they should be more doable.
Additionally, here are some online notes on change of variables for double integrals:
Polar Coordinates
Change of Variables
May 3:
Updated Double Integrals/Change of Variables worksheet.
May 26:
Midterm Solutions.
Final Exam Review.
And remember! Final drafts for the challenge problem are due on Wednesday, the last day of class. I also made a correction to problem 5 in the challenge problem list (the integral problem, changing a minus to a plus), so check it out if you've been struggling with that one. Props to those who caught it!
June 1:
Final Exam Review Solutions.
Make-up class notes on random variables.
Date
|
Day
|
Section
|
Problems
|
3/28
|
M
|
10.1-2
|
10.1: 4, 5, 6, 12, 14, 17
10.2: 15, 16, 17
|
3/30
|
W
|
10.2
|
7, 8, 22, 23, 24, 27, 28, 33
|
4/1
|
F
|
10.3
|
7, 8, 9, 22, 23, 29, 42, 46
|
4/4
|
M
|
10.4
|
7, 8, 9, 16, 27, 35, 37
|
4/6
|
W
|
10.5
|
1, 3, 4, 5
|
4/8
|
F(Q)
|
10.5
|
17, 18, 23, 26, 28, 40, 44
|
4/13
|
W
|
10.6
|
3, 4, 7, 10, 17, 28
|
4/14
|
Th
|
|
(lumped in with 4/15)
|
4/15
|
F
|
10.6
|
21,22,23,25,35,36,37,42,61
|
4/18
|
M
|
Online
Notes
|
Worksheet,
#5-14
|
4/20
|
W
|
Online
Notes
|
Worksheet,
#5-14
|
4/22
|
F(Q)
|
Online
Notes
|
Change of Variables
Worksheet
|
4/25
|
M
|
Online
Notes
|
Change of Variables Worksheet
|
4/27
|
W
|
11.1
|
1, 3, 19, 20, 21
|
4/29
|
F(M)
|
Midterm
|
|
5/2
|
M
|
11.1
|
19, 21, 23, 24, 25
|
5/4
|
W
|
11.1
|
(see friday)
|
5/6
|
F(Q)
|
11.1/Euler's Formula
|
5, 6, 9, 10, 11, 27, 28, 35, 43,
44
Worksheet
|
5/9
|
M
|
|
|
5/11
|
W
|
11.2
|
1, 2, 9, 12, 15, 21, 22
|
5/13
|
F
|
11.2
|
23, 24, 27, 28
|
5/16
|
M
|
11.3
|
7, 8, 9, 10, 13, 14
|
5/18
|
W
|
11.4
|
17, 18, 19, 20
|
5/20
|
F(Q)
|
12.1
|
5,6,25,27,29,32,37,43,44
|
5/23
|
M
|
12.2
|
20,28,41,42,50,51
|
5/25
|
W
|
12.2
|
13-18,26,27,48,49
|
5/27
|
F
|
12.3
|
2,3,7,8,9,15,16,22
|
5/30
|
M
|
Memorial Day
|
|
6/1
|
W(Q)
|
Last Day of
Class
|
Challenge Problem Due
|
|
|
|
|
Philosophy
Cooperation
Studies and experience have shown that collaboration leads to
better learning. Working with friends (or enemies) provide more eyes on
a problem, and a greater likelyhood that someone in the group
understands a subtle point. Additionally, the act of teaching concepts
greatly increases one's understanding of those concepts: Even a star
student will gain a lot from working with a group.
That said, it is expected that all work handed in will be written up by
individuals, in their own words. This is because, at the end of the
day, I need to be able to evaluate the quality of each person's work
individually. More importantly, the process of writing down mathematics
and expressing ideas is fundamental to te development of understanding.
Problem Solving
Mathematics is the art of solving problems using logical structures.
There are two kinds of mathematics courses: First, technique-oriented
courses (often with names like `Calculus') teach specific mathematical
methods for solving certain specific kinds of math problems. The second
kind of mathematics course is problem-oriented, rather than
technique-oriented. This kind of course aims to improve student's
skills with attacking problems through the creative use of mathematics.
This kind of course has traditionally been used to prepare students for
math competitions, but sometimes introduction to proof courses are
problem-oriented as well.
Likewise, assignments in math classes fall into two broad
categories. The first, and more common, type tests the student directly
on what they've just learned in the present section, usually applying a
specific technique. The second type consists of states a more-or-less
open ended question, presented without a clear means of attack. Paul
Zeitz, in his excellent book, `The Art and Craft of Problem Solving,'
calls these two categories exercises and problems, respectively.
Calculus is primarily taught as a techniques class with lots of
exercises and few if any real problems. However, problem solving is
essential to understanding how to apply mathematics. People often ask
math teachers of their classes, `When will I ever use this?' But to get
the most out of the class, one should instead ask oneself of the world,
`Where will I apply this?'
Luckily calculus is very, very applicable. Pretty much anywhere you
encounter questions about change or accumulation, and can devise some
method of measurement, one can build an application of calculus. In
fact, calculus -- like many of the most robust areas of mathematics --
was built specifically in response to problems involving these
concepts. Mathematics broke free of the static geometries of Euclid and
its applications in architecture and evolved into a language capable of
describing the ever-changing realities of the world. In this regard,
calculus and differential equations has been extremely successful.
Problems are inherently harder to solve than exercises, and also
harder to grade. Nonetheless, I would be remiss if I didn't integrate
some kind of real problem solving into the course, which is what the
problem write-up is intended to accomplish. A variety of problems will
be provided, with the hope that everyone will find something that they
can crack.