Syllabus |
Homework assignments |
About the exams |
Using your computer |
Lecture schedule
Each week you will find a homework assignment here. It will always
consist of a reading assignment and a list of problems to do. You
should be doing these problems on your own, or, better with a group
of friends. In discussion your TA can help you with the problems
that cause the most headaches. You can
use the
piazza site for questions: keep checking—maybe someone
asked a question you’re stuck with, or one you know how to
answer, or maybe someone asked something you didn’t know you
didn’t know.
The bi–weekly quizzes will be based on the homework and
reading assignments.
On Wednesday we covered the material in sections 1—6, Chapter 1 of the text.
We will cover the second half of Chapter 1 on Friday.
Reading: Chapter 1. Problems: 1—15 from Chapter 1, and problems 1 and 2 from Chapter 2.
In questions that were asked on piazza, the following two clippings from the
math 222 text appeared.
About position vectors:
About normals to a plane, the cross product, and the equation of that
plane:
September 9, 11
We reviewed the cross product and started on vector functions (chapter 2).
After having seen linear and circular motion we started on the cycloid example, and
also the helix. We started discussion of the derivatives of vector functions.
Reading: Chapter 2.
Problems: All problems from Chapter 2. Until Monday’s lecture,
when I will explain tangent, unit tangent, acceleration vector and arc length,
these problems will be hard to do. So instead of trying to do the problems, read
the problems, then read the text: read chapter 2! Which sections have been covered
in lecture already? Which sections look like they might be useful in the problems?
Extra problems on lines:Here are some problems on
parametric representation of lines with answers.
About problem 4.
Here is a short animation of the curves and lines that appear in Problem 4 of
Chapter 2. The animation shows
The curve $\vec x(t) = \langle t, t^2, t^3 \rangle$
A point on the curve (red dot)
The tangent to the curve at the red dot (white line)
The $xy$–plane
The intersection point of the tangent with the $xy$—plane
What happens to all these objects if you change the original point on the curve
And here is an animation of how the cycloid comes about:
September 14, 16, 18
We finished chapter 2
on curves and vector functions and started chapter 3 on functions of
several variables. On Monday and Wednesday next week, we will see
quadratic functions of two variables and how to classify them as
positive, negative, or indefinite, using good old “completing the square.”
Reading: Chapter 3, section 1, 2, and 3.
Problems: Chapter 3 (p.46) 1—7, 9, 10.
September 21, 23, 25
We finished Chapter 3 and started on chapter 4 (partial derivatives).
Reading: Chapter 3 and Chapter 4, §1–2.
Problems: Prepare for the midterm next week,
see the exam page
September 28, 30, October 2
The midterm was on Wednesday. On Monday and Friday we saw the linear approximation
formula, the equation for the tangent plane, and the chain rule that tells us how
to differentiate $\frac{df(x(t), y(t))}{dt}$.
Reading: Chapter 4: §2 has a detailed explanation of the linear
approximation formula. §4 explains what the error term is—read it to see
what we are ignoring most of the time. §5 is about tangent planes and §6
presents the chain rule.
Problems: Chapter 3 (p.48) 14 (pick a
few; the desmos
grapher has answers), 16, 17.
Chapter 4, §3, problems 1, 2 (pick a few, do more later). See
using your computer for a demo on
how to use the Google grapher to
make 3D graphs.
This week we met the gradient
of a function, which lets you write the chain rule in vector form.
Reading: Sections 8, 9, and 10 from chapter 4.
Problems: Right after §10, starting on page 72:
problem 1–do a few.
problem 2–pick one
problems 3—11.
October 12, 14, 16
Chain rule for functions of several variables, especially, how to compute
the derivatives of expressions like $f(x(u, v), y(u,v))$.
We saw Clairaut’s theorem which says that mixed partials
don’t depend on the order of the derivatives: $f_{xy}=f_{yx}$.
Related: when can you, and how do you find a function $f(x,y)$ if
you are given its derivatives $f_x$ and $f_y$?
Friday’s lecture showed you a derivation of the “wave
equation” which describes a vibrating string (and many, many
more things). Here are the slides and
here are the graphs on the desmos website.
Reading: Section 11 from chapter 4, in particular examples 11.1 and 11.3.
§15 : 12—14 (about recovering a function from
its derivatives). It turns out that problem 14 is a
duplicate of problem 12. Instead, do
this similar problem.
October 19, 21, 23
We started Chapter 5 on
Optimization. Definitions of local/global maximum and minimum,
interior and boundary point of a region; def of “closed
region”. Main theorems in §2.1 and §4.2. The
“fishy example.” On Friday we ended by looking at
linear regression.
Reading: Chapter 5, §1—7.
Problems: from Chapter 5.
§3 : 1—3. (note the “fishy
example” done in lecture differs from the one in the
problems by a minus sign.)
§6: 1– pick a few, save some for later.
§6: 2, 3, 4, 5, 6, 7.
October 26, 28, 30
On Monday we saw Taylor’s formula for two variables, on Friday we
applied this to the second derivative test. The example we saw was the
function $f(x,y) = x^3+y^3-3xy$ (we didn’t discuss the zero set,
but it is a “famous curve” called the
Folium of Descartes. See also wikipedia)
Reading: Chapter 5, §9 covers the Taylor expansion
and the 2nd derivative test. In the coming week we will be seeing the
method of Lagrange multipliers, which is presented in §12.
Problems: all from Chapter 5, § 10:
1 (to make sure you understood §9)
2a, c, e.
5a, c, e, h, k, o.
6, 8, 9.
This section has many more problems. Keep those for midterm practice.
November 2, 4, 6
We saw Lagrange’s method for finding maxima and minima when their
are constraints. On Wednesday we went through an example from
economics involving the
Cobb—Douglas production function. On Friday we started on
double integrals.
Reading:
For Lagrange multipliers make sure you understand Figure 8 on page 102.
For double integrals read §1 and §2 of Chapter 6. In particular
look at figures 3 and 4 on page 111 and compare with figure 1 on page 107.
Double integrals, iterated integrals, integrals in polar coordinates.
We finished with a computation of the area under the “Bell curve”
(you can find the whole story on Wikipedia; see “computation by polar
coordinates” on this page for the
derivation I gave).
Reading:
Finish Chapter 6 on integrals. In particular,
§5 has a number of interpretation of multiple integrals (the
“average of a function over a region”, “mass is integral of
density”, “moment of inertia.”
§6 shows how to compute integrals using cylindrical and spherical
coordinates. Figures 15 and 16 are important: you should be able to
reproduce them.
Problems:
Double integrals: Ch.6, §3: 5, 6 (pick a few do the rest before the
midterm).
Ch 6, §3: 7, 8 (8 has a typo: instead of $a\le c\le b$ it should say
$a\le x\le b$).
Triple integrals: Ch.6, §7: 1—3, 5, 6.
Ch 6, §7: 7a,c,e; 8, 10 (read the explanation right above that problem),
14, 17.
November 16, 18, 20
We saw how to set up a triple integral that computes the kinetic energy
of a rotating body and examples of how to compute such triple integrals
in cylindrical coordinates and spherical coordinates.
Reading:
Review previous reading about double integrals, triple integrals, and
cylindrical and spherical coordinates. Then begin the last chapter,
Chapter 7 on Vector fields and line integrals. In particular,
§1—2 which give examples of vector fields.
§5 has a number of interpretation of multiple integrals (the
“average of a function over a region”, “mass is integral of
density”, “moment of inertia.”
§6 shows how to compute integrals using cylindrical and spherical
coordinates. Figures 15 and 16 are important: you should be able to
reproduce them.
