For Fri, Sep 24:

For Mon, Sep 27:

For Wed, Sep 29:

For Fri, Oct 1:

For Mon, Oct 4:

For Wed, Oct 6:

For Fri, Oct 8:

For Mon, Oct 11

For Wed, Oct 13

For Fri, Oct 15

For Mon, Oct 18

For Wed, Oct 20

there will be no homework to be turned in.

For Fri, Oct 22

For Mon, Oct 25

For Wed, Oct 27

for Thurs, Oct 28

for Mon, Nov 1

for Wed, Nov 3

for Fri, Nov 5

for Mon, Nov 8

for Wed, Nov 10 Study for the exam. Do the practice exam, you will not have to turn this in.

Here a few corrections.

Number 4: the second plane should be

A2 x + B2 y + C2 z + D2 =0,


A1 x + B1 y + C1 z + D1 =0.

Number 8: the point D should be D = (1,-1), not (-1,1).

Number 6: first note that by the set of cubic polynomials it is meant all polynomilas in the form a3 x3 + a2 x2 + a1 x + a0 where the ai are real numbers. ( i.e. we are allowing that any of the ai are zero. So for example, for the purposes of this problem a quadradic polynomial is a degenerate example of a cubic polynomia.) Secondly, to do the the problem you are required to chose a range of T there are several possible choices. You may fail to be able to do part (c) if you don't make a finite dimensional choice, so this you should do.

for Fri, Nov 12

for Mon, Nov 15

for Wed, Nov 17

for Fri, Nov 19

Note to Prof. Webb's students: on Friday, you are required to turn in the assignment listed above as being due on Wednesday - not the assignment listed below.

for Mon, Nov 22

for Mon, Nov 29

for Wed, Dec 1

for Sunday Dec 4 Try this prictice exam (it is of course not due.)

Math 9 Practice Final

Find the Taylor series about x=0 of


What is the radius of convergence of this series?

Find an equation for the tangent plane to the graph of f(x,y) = xy + y2 + e2x at (0,2,5).

Find an equation for plane parallel to this tangent plane containing the point (1,1,0).

Where does the line perpendicular to this new plane and containing (1,1,0) hit the original tangent plane.

Let f(x,y) = xy and $g(x,y) = e^y \sin(x)$. Describe the points where the level sets of f and g are perpendicular.

Find the minimum and maximum values of f(x,y) = (x + y)e-(x2 + y2) on the disk $x^2 + y^2 \leq 1 $.

Let $f(x,y,z) = ((yz) \sin(xy) , (yz)e^{(xz)} + (xy)) $.
Express f as a composite of a function $g: R^3 \rightarrow R^2$ and a function $f: R^3 \rightarrow R^3$.
Find Df and Dg (Df refers to the derivative of f).
Use the chain rule to compute $Df(\frac{\pi}{2},1,1)$.
Find the kernel of $Df(\frac{\pi}{2},1,1)$.


\begin{displaymath}M_{\lambda} = \left[\begin{array}{rr} \lambda & 0 \\ 0 & 1 \end{array}

is a stretch of the plane by $\lambda$ in the direction of the x-axis. Let $M^{\theta}_{\lambda}$ denote the stretch by $\lambda$ in the direction determined by the line inclined at an angle $\theta$ with respect to the x-axis. Find a formula for $M_{\lambda}^{\theta}$.


\begin{displaymath}I + (\lambda -1) \left[\begin{array}{rr} (\cos(\theta))^2 &
...(\theta) \sin(\theta) &
\end{array} \right] .\end{displaymath}

Let V be the vector space of polynomials of degree $\leq 3$ and let A denote the linear transformation of V to V given by

\begin{displaymath}A(p(x)) = a \frac{d^2p}{dx^2} + b \frac{dp }{dx} + c p .\end{displaymath}

Express A as a matrix.

In each of the following cases describe the rank of A, and find a non-zero polynomial in A's kernel whenever such an element exists.
$a \neq 0$ and $b \neq 0$ and $c \neq 0$.
$a \neq 0$ and $b \neq 0$ and c = 0.
$a \neq 0$ and b = 0 and c = 0.

answer to part (a) if we let x3 corrspond to the first standard basis vector, x2 the second, x the third, and 1 the fourth the anser to a is

\begin{displaymath}A = \left[\begin{array}{rrrr} c & 0 & 0 & 0 \\ 3 b & c & 0& 0 \\
6a & 2b & c & 0 \\ 0 & 2a & b & c
\end{array} \right] .\end{displaymath}

Find the general solution to

\begin{displaymath}\frac{d^2f}{dx^2} - 3 \frac{df }{dx} + 2 f= 0 . \end{displaymath}

Use the previous problem (problem 7(a)) to find a polynomial solution to

\begin{displaymath}\frac{d^2f}{dx^2} - 3 \frac{df }{dx} + 2 f = x^3 + 3x +1. \end{displaymath}

Find a solution to

\begin{displaymath}\frac{d^2f}{dx^2} - 3 \frac{df }{dx} + 2 f = x^3 + 3x +1\end{displaymath}

satisfying f(0)=1 and $\frac{df}{dx}(0) = 2$.

answers (a)

f(x) = a e2x + b ex


\begin{displaymath}p(x) = \frac{1}{8} (4x^3 + 18
- 54 x -95)\end{displaymath}


\begin{displaymath}f(x) = p(x) - \frac{33}{8}e^{2x} + \frac{136}{3} e^x \end{displaymath}

Math 9 Fall 1999