
\input graphicx
\nopagenumbers
\magnification 1400
\vsize 10truein
\voffset -.4truein
\parskip 15pt
\parindent 0pt
\hsize 7.5truein
\voffset -.7truein
%\hsize 10.2truein
\hoffset -.4truein
\def\R{{\bf R}}
\def\v{{\vskip 20pt}}
\def\hb{{\hfil \break}}

\def\hr{{\vskip 10pt \hrule}}

Quiz 6 Form A \hfil \break
Nov 10, 2017

[10]~~ 1.  ${\cal L}(u_2(t)(t + 5)) =  \underline{\hskip 2in}$

\v\v

[10]~~ 2.  Find the eigenvalues and corresponding eigenvectors for $ A = \left(\matrix{1 & 2 \cr 2 & 4}\right)$ 

An eigenvalue of $A$ is    $\underline{\hskip 1in}$. The eigenvectors corresponding to this eigenvalue

\v

\rightline{ are all nonzero multiples of  $\underline{\hskip 1in}$.}

An 2nd eigenvalue of $A$ is    $\underline{\hskip 1in}$. The eigenvectors corresponding to this \v

\rightline{ eigenvalue are all nonzero multiples of  $\underline{\hskip 1in}$.}
\end

\vskip 10pt

%\centerline
{$W(t) = \underline{\hskip 1in}$}

\v\v
\vfill
%$p(t) = {4 \over t}$.  Thus by Abel's thm $W(t) =ce^{-\int  {4 dt \over t}}= ce^{-4\int  {dt \over t}}= ce^{4 ln|t|} =e^{ln|t|^4} = ct^4$ .

%Thus  $W(t) =ct^4 $ for some constant $c$.  Since $W(1) = 3$,   ~~$3 =  c(1)^4$.  Thus $c = 3$.

%Hence $W(t) =3t^4 $


[10]~~2.)  Write $y = \sqrt{3}cos(5t) - sin(5t)$ in the form $y = Rcos(\omega t - \delta)$.  Determine the period, phase, and amplitude.

$y $$=  \underline{\hskip 3in}$ 
\vskip 10pt

period$=  \underline{\hskip 1in}$, phase$=  \underline{\hskip 1in}$, and amplitude$=  \underline{\hskip 1in}$.

\v
%$Rcos(\omega t - \delta) = Rcos(\delta) R(\omega t) + Rsin(\delta)Rsin(\omega t) =  \sqrt{3}cos(5t) - sin(5t)$

%Thus $\omega = 5$ and $ Rcos(\delta) =  \sqrt{3}$, $Rsin(\delta) =  - 1$

%Thus $tan(\delta) = {R cos(\delta) \over Rsin(\delta) } = - \sqrt{3}$.

%Since  $cos(\delta) > 0$ and $sin(\delta) < 0$, $\delta$ is in quadrant 4 ( i.e, $-{\pi \over 2} < \delta < 0)$.  Thus $\delta = -{\pi \over 3}$  

\vfill
\end