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\noindent Homework 11

\noindent Math 336, Winter `00

\noindent Due Monday, 5/1.
\vskip .2in

\noindent Solve the last three problems on the second exam.
\bigskip

\noindent{From Wade's book:}
\begin{itemize} 
\item Page 358: 1bd, 6
\end{itemize}  

\noindent{\bf Other Problems}
\bigskip

\noindent{\bf A.}  
          In problem 1b, use the method described in class to approximate
          a solution $(x,y)$ of $f(x,y) = (.1,1.2)$.  Then use your
          first approximation to generate a second approximation.
\bigskip 

\noindent{\bf B.} This problem is designed to show what can go wrong when 
the hypotheses of the inverse function theorem fail.
\begin{enumerate} 
\item Define $f:\R^2\to\R^2$ by $f(x,y) = (x^2 + y^2, x-2y)$.  
\item At which points $(x,y)\in\R$ are the hypotheses of the inverse
      function theorem satisfied?
\item Pick any point (your choice---I want you to use actual numbers here) 
      $(x_0,y_0)\in\R^2$ where the hypthoses 
      of the inverse function theorem fail, and let 
      $(r_0,s_0) = f(x_0,y_0)$.  Pick a number $r$ slightly larger than
      $r_0$.  How many real solutions $(x,y)$ of $f(x,y) = (r,s_0)$ are there
      near $(x_0,y_0)$?  What if $r$ is slightly less than $r_0$?  
\item Explain why your answers to the last part rule out the existence of
      a local inverse for $f$.  Explain what's going on here in geometric 
      terms.
\end{enumerate}

\noindent{\bf C.}  Inverse functions can exist even when the hypotheses of 
the inverse function theorem aren't satisfied.  Consider $f:\R\to\R$.
At what points $x\in\R$ do the hypotheses of the inverse function theorem
fail?  Show that nevertheless, there exists a function $g:\R\to\R$ such
that $g\circ f(x) = x$ for every $x\in\R$.  What is the difference 
between this $g$ and the one guaranteed in the conclusion of the inverse
function theorem?



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