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{\bf Math 9 Practice Final}

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\begin{enumerate}




\item
\begin{enumerate}
\item
Find the Taylor series about $x=0$ of 
\[\arctan\left(\frac{x}{2}\right).\]
\item
What is the radius of convergence of this series?
\end{enumerate}




\item
\begin{enumerate}
\item
Find an equation for the tangent plane to the graph of 
$f(x,y) = xy + y^2 + e^{2x}$ at $(0,2,5)$.

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

\item
Where does the line perpendicular to this new plane and containing
$(1,1,0)$ hit the original tangent plane.
\end{enumerate}

\item
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.  

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


\item
Let $f(x,y,z) = ((yz) \sin(xy) , (yz)e^{(xz)} + (xy)) $.

Note: If you have already tried this the point was meant to be
\[(\frac{\pi}{2},0 ,0) \] 
not 
\[\[ (\frac{\pi}{2},1,1)\]
 as stated.  Both ways
are 
doable but this second point is much easier.

\begin{enumerate}
\item
Express $f$ as a composite  of a function $g: R^3 \rightarrow R^2$  and
a function $h: R^3 \rightarrow R^3$. 
\item
Find $Dh$ and $Dg$ ($Df$ refers to the derivative of $f$).
\item
Use the chain rule to compute $Df(\frac{\pi}{2},1,1)$.
\item
Find the kernel of    $Df(\frac{\pi}{2},1,1)$.
\end{enumerate}

\item
Notice  
\[ M_{\lambda} = \left[\begin{array}{rr} \lambda & 0 \\ 0 & 1 \end{array} 
\right]
\]
is a stretch of the plane by $\lambda$ in the direction of the
x-axis leaving the orthoganal direction fixed.  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 with its orthogonal direction
fixed.  Find a formula for 
$M_{\lambda}^{\theta}$.  



\item
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
\[ A(p(x)) = a \frac{d^2p}{dx^2} + b \frac{dp }{dx} + c p .\]
\begin{enumerate}
\item
Express $A$ as a matrix.

\item
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.
\begin{enumerate}
\item
$a \neq 0$ and $b \neq 0$ and $c \neq 0$.
\item
$a \neq 0$ and $b \neq 0$ and $c =  0$.
\item
$a \neq 0$ and $b = 0$ and $c =  0$.
\end{enumerate}  
\end{enumerate}

\item 
\begin{enumerate}
\item
Find the general  solution to 
\[ \frac{d^2f}{dx^2} - 3  \frac{df }{dx} + 2 f= 0 . \]
\item
Use the previous problem (problem 7(a)) to find a polynomial solution to

\[  \frac{d^2f}{dx^2} - 3  \frac{df }{dx} + 2 f = x^3 + 3x +1. \]

\item
Find a solution to 

\[  \frac{d^2f}{dx^2} - 3  \frac{df }{dx} + 2 f = x^3 + 3x +1 \]
satisfying 
$f(0)=1$ and $\frac{df}{dx}(0) = 2$.
Note: you are the right track if you get coeffeicients wich are rational
numbers with denominators like 8.
\end{enumerate}



\end{enumerate}




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