M408M, Multivariable Calculus

This page is always under construction, so
you
should check it regularly.

The written assignments will appear on this page. Some just involve doing book problems. Others are worksheets. The worksheets that are handed out in class (either lecture or discussion) will later appear here in PDF form, but not necessarily immediately.

All written assignments based on the book
are to be turned in at the **beginning** of
discussion section, usually on a Monday.
Worksheets will sometimes be due at the end of a discussion section
(if the main activity for the discussion is working on the worksheet)
and sometimes at the beginning (if you mostly worked it on a previous day
and were allowed to finish at home).

Online homework is always on Quest. There will be a pre-class and a post-class assignment for each class meeting. The pre-class assignment is due at midnight the night before the class. The post-class assignment is due at 6PM on the day before the subsequent class. In other words, the pre-class assignment for Tuesday is due Monday night, and the post-class is due Wednesday afternoon; the pre-class assignment for Thursday is due Wednesday night and the post-class is due on Monday afternoon.

The class schedule, and hence the schedule of online homework, can be found here.

**Homework #1**, due Wednesday, September 9

Section 10.1 page 642, problems 24, 25, 26, 37.

Section 10.2 page 651, problems 29, 32.

Section 10.3, pages 662-3, problems 16, 20, 26, 62.

**Homework #2**, due Wednesday, September 16

Section 10.4, page 668, problems 8, 26, 45, 46.

Section 10.5, page 676, problems 9, 16, 18, 51.

Section 10.6, page 684, problems 5, 8, 12, 16, 29.

**Homework #3**, due Monday, September 21

Section 12.1, page 790, problems 6, 10, 12, 16

Section 12.2, page 798, problems 2, 8, 30, 38

Section 12.3, page 807, problems 40, 46, 48.

**Homework #4**, due Monday, September 28

Section 12.4, page 814, problems 30, 36, 38, 48

Let $Q(3, -2, 7)$ be a point and let $L$ be the line whose
equation in vector form is $\langle x,y,z \rangle = \langle
-2, 5, -3\rangle + t \langle 1, 2, -2 \rangle$.

a) Find the equation of the plane $P_1$ that contains $Q$ and $L$.

b) Find the equation of the plane $P_2$ that contains $Q$ and is
orthogonal to $L$.

c) Find the equation of a line through $Q$ that is orthogonal to $P_2$.

d) Find the distance from $Q$ to $L$.

e) Find the equation of a line through $Q$ that intersects $L$.
There are many possible answers (depending on where it hits $L$).

f) Find the equation of a line through $Q$ that does NOT intersect $L$.
How do you know that the two lines don't intersect?

Section 12.5, pages 824-6, problems 22, 50, 68

**Homework #5**, due Monday, October 5

Section 12.6, pages 832-4, problems 10, 21-28, 49

**Homework #6**, due Monday, October 12

Section 13.1, pages problems 15, 16, 21-26

Section 13.2, problem 52.

There's only one written problem from Section 13.2. None of the
other ones in the book looked very instructive.

**Homework #7**, due Monday, October 19

Section 13.3, problems 16, 38 and 56.

Section 13.4, problems 2, 6, 16, 36, 38, 42.

**Homework #8**, due Monday, October 26

Section 14.1, problems 5, 7, 8, 38

Section 14.2, problems 4, 14

Section 14.3, problems 4, 6, 10, 96.

**Homework #9**, due **Wednesday**, November 4

Section 14.4, problems 12, 22, 23, 32, 33, 42

Section 14.5, problems 36, 44, 46

**Homework #10**, due Monday, November 16

Problem 1)
Download a map of Enchanted Rock State Natural Area at
http://www.tpwd.state.tx.us/publications/pwdpubs/media/park_maps/pwd_mp_p4507_119c.pdf and print out 3 copies. On one copy, sketch the gradient of the
height function.
On a second copy, mark points where the gradient is unusually large. How
would you describe the terrain near those points? On the third copy, mark
points where the gradient is practically zero. What is happening near those
points? Which ones are local maxima? Local minima? Saddles?

Problem 2) Suppose we have a line $y=mx+b$ that we are using to find a
linear fit to the points $P(1,1)$, $Q(2,2)$ and $R(3,4)$ in the
$x$-$y$ plane. Let $\epsilon_1$ be the amount that $P$ lies above the
line (which may be negative if $P$ is actually below the line),
$\epsilon_2$ be the amount the $Q$ lies above the line, and
$\epsilon_3$ be the amount that $R$ lies above the line.

(a) Write formulas for $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ as
functions of $m$ and $b$.

(b) Let $E(m,b) = \epsilon_1^2 + \epsilon_2^2 + \epsilon_3^2$ be
the total squared error. We want to find the value of $(m,b)$ that
minimizes $E$. (Remember that $E$ is a function of $m$ and $b$, not a
function of $x$ and $y$!) This involves taking a gradient and setting
it equal to zero. Write down the gradient $\nabla E(m,b)$. Simplify as
much as possible.

(c) Now solve the equations $\nabla E(m,b) = 0$ to figure out what the
best values of $m$ and $b$ are. So what is the equation for the best
line through the points $P$, $Q$, and $R$?

(d) Using the best
line you found in (c), compute $\epsilon_1$, $\epsilon_2$ and
$\epsilon_3$. Does you consider this line a good fit or a bad fit?
Explain.

Section 14.6, problems 32, 34, 44

**Homework #11**, Due Monday, November 23.

Find the extremal values of $f(x,y,z) = 2xy + 3z^2$ on the
surface $x^2+y^2+z^2=1$. You don't need to know any linear algebra to
do this, but what you are doing is equivalent to finding the
eigenvalues and eigenvectors of the matrix $\begin{pmatrix} 0&1&0 \cr
1&0&0 \cr 0&0&3 \end{pmatrix}$.

Stewart, Section 14.7, problems 4, 16, 40, 50

Stewart, Section 14.8, problems 14, 30, 40

Stewart Section 15.2, problems 6, 8, 12, 38

**Homework #12**, Due **Wednesday**, December 2.

Stewart Section 15.3, problems 26, 50, 54, 56.

Stewart Section 15.4, problems 24, 28a (skip b), 36

**Practice Homework**. This assignment is **not** supposed to
be turned in, and is intended as practice for the final exam.

Stewart Section 15.5, problems 1, 3, 7, 11, 17.

Stewart Section 15.10, problems 1, 5, 9, 10, 15, 23.

Note that these are all odd numbered problems, so there are solutions in the
back of the book.