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, Friday, and
Monday we will cover the material in Chapter 1 of the text.
Reading: Chapter 1. Problems: 1—15 from Chapter 1.
The following two clippings from the part of the math 222 text
about vectors may be useful:
About position vectors:
About normals to a plane, the cross product, and the
equation of that plane:
September 13–22
We started on vector
functions (chapter 2), linear and circular motion, and also the
helix. We then looked at derivatives of vector functions and
interpreted the derivative as instantaneous velocity of a
motion, as well as the direction of the tangent to the curve traced
out by a vector function.
Reading: Chapter 2.
Problems from §17, chapter 2:
1, 2, 3cd (3ab will be done in class on Monday), 4, 5, 6, 7;
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
We started chapter 3 on
functions of several variables, defining their graph, and
determining the domain of a number of functions. On Monday 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.” We will discuss level sets of
functions, functions in polar coordinates, and “functions as
movies”.
Reading: Chapter 3.
Problems: Chapter 3 (p.46) 1—17, skipping problem 13.
October 9—13
We finished Chapter 3 and started on chapter 4 (partial derivatives).
Reading: Chapter 3 and Chapter 4, §1–2.
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 meet 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:
Chapter 4, §7 problems 4, 5.
Right after §10, page 72: problems 1, 2 ( $\vec \nabla
(fg)$ ), 3—11.
October 30 – November 3
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 $f_y$?
Reading:
The material on second derivatives and the
method for finding a function from its first order partials is in
Sections 13 and 14 from chapter 4.
The material covered this week is in sections 1–9 of chapter 5.
§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.
§15 : problem 15 to practice the chain rule with second
order derivatives
From chapter 5:
§3 : problem 1, 2.
§6: 1a–1r — pick a few, save some for later.
§6: 2, 3, 4, 5, 6, 7.
November 6—10
Last week 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.” in §4.4. In
§4.5 you find another similar example. This week we will see
the second derivative test for critical points, and go on to the
method of Lagrange multipliers.
Reading: Chapter 5, §1—6; §9.
Problems: from Chapter 5.
Finish&review last week’s homework. Several of
the new problems are continuations of last week’s
problems.
§10: 1
(asks you to read the text), 2abcd
,
3efgh (quadratic forms again, but now the variables
are called $\Delta x$ and $\Delta y$), 5aehik
, 5opqr (these are examples where the second
derivative test is inconclusive; verify this, and also take
a look at the graphs of the functions, either by drawing
them yourself, or by checking on google) 6, 7bc, 8.
November 11—17
Last week saw Taylor’s formula for two variables and applied it to the the second derivative test. The example $f(x, y) = x^2-x^3-y^2$. One other example we saw was the
function $f(x,y) = x^3+y^3-3xy$ (I drew the zero set without explaining why it looks like that; it is a “famous curve” called the Folium of Descartes. See also wikipedia).
For Lagrange multipliers and Optimization with Constraints see Chapter 5, §12.
In the coming week we will start on multiple 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.
Problems:
Chapter 5, § 10, do the remaining problems.
Chapter 5, § 13: 1, 2, 3, 6, 7, 8, 10, 11, 12.
November 27—December 1
Double integrals, iterated integrals, integrals in polar coordinates.
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: 1, 2, 3, 4, 5, 6 (pick a few do the rest before the final), 7.
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.
