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\begin{center}{ \Large{Mathematica practice}  \\\large{9/19/2017} }\end{center}
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% into an ``enumerate'' environment for the problem statements.
\bigskip
\textbf{Matrix multiplication}\\\\
Recall: to introduce a matrix, say \[
   M=
  \left[ {\begin{array}{cc}
   1 & 2 \\
   3 & 4 \\
  \end{array} } \right]
\]
write $M=\{\{1,2\},\{3,4\}\}$. To multiply two matrices $A$ and $B$, type $A.B$\\

\emph{Practice}\\ Introduce the matrices ($v$ is a vector, but can be regarded as a matrix)
\[
   A=
  \left[ {\begin{array}{ccc}
   1 & 2 &3\\
   4 & 5 &6\\
   7 & 8 &9\\
  \end{array} } \right],~~~~B=
  \left[ {\begin{array}{ccc}
   1 & 0 &1\\
   2 & 3 &1\\
   0 & 1 &-1\\
  \end{array} } \right],~~~~C=
  \left[ {\begin{array}{cc}
   1 & -1\\
   0 & 1 \\
   2 & 0 \\
  \end{array} } \right],~~~~v=
  \left[ {\begin{array}{c}
   1\\
   2 \\
   3 \\
  \end{array} } \right]
\]
Now try A.B, A.C, C.A, A.v, v.A.\\Next, compute the determinants of A and B and A.B. Is det(A)det(B) = det(AB)?
\vspace{.1in}
\hrule
\vspace{.1in}
\textbf{Solve an equation}\\\\
\begin{tabular}{ll}
$f(x)=0$ &  Solve[f(x)==0,x]\\
$f(x)=0$ in the interval (a,b) & Solve[f(x)==0 \&\& a$<$x$<$b,x]\\
$f(x,y)=0$ & Solve[f(x,y)==0,\{x,y\}]\\
system $f(x,y)=0$ and $g(x,y)=0$ & Solve[f(x,y)==0 \&\& g(x,y)==0,\{x,y\}]\\
\end{tabular}\\
\linebreak\linebreak
\emph{Practice}\\
Solve the following equations\begin{enumerate}[i)]
\item $x^3-3x+1=0$
\item $\sin x=2\cos x$ ~for $x\in[0,3\pi]$
\item \[
\left\{ 
\begin{array}{c}
x+z=1 \\ 
2x+3y+z=2 \\ 
y-z=3
\end{array}
\right. 
\]
\end{enumerate}
\vspace{.1in}
\hrule
\vspace{.1in}
\textbf{Differentiation}\\\\
\begin{tabular}{ll}
$f'(x)$ &  D[f(x),x]\\
$f'''(x)$  & D[f(x),\{x,3\}]\\
$\dfrac{\partial f}{\partial x}$ of a function f(x,y) & D[f(x,y),x]\\
$\dfrac{\partial^3 f}{\partial x^2\partial y}$ of a function f(x,y) & D[f(x,y),x,x,y]\\
\end{tabular}\\
\linebreak\linebreak
\emph{Practice}\\
Given $f(x,y,z)=xy^{50}+x^{10}y^{15}z^{20}$. Compute
\begin{enumerate}[i)]
\item $\dfrac{\partial f}{\partial y}$
\item $\dfrac{\partial^2 f}{\partial x\partial y}$
\item $\dfrac{\partial^{29} f}{\partial x^3\partial y^{12}\partial z^{14}}$
\end{enumerate}
\vspace{.1in}
\hrule
\vspace{.1in}
\textbf{Limit}\\\\
\begin{tabular}{ll}
$ \lim_{x \to a} f(x) $ &  Limit[f(x), x$\to$a]\\
$ \lim_{x \to a} f(x,y) $ &  Limit[f(x,y), x$\to$a]\\
$ \lim_{x=2y,y \to a} f(x,y) $ (limit in the direction $x=2y$) &  Limit[f(x,y)/.x$\to$2y, y$\to$a]\\
\end{tabular}\\
\linebreak\linebreak
\emph{Practice}\\
Compute
\begin{enumerate}[i)]
\item $f'(1)$ where $f(x)=\sin x\cos x $
\item $\dfrac{\partial f}{\partial x}(1,2)$ where $f(x,y)=x^2y$
\end{enumerate}
\end{document}













