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\usepackage[all]{xy}
\usepackage{color}
\usepackage{epsfig}
\usepackage{graphicx}
\def\dsp{\displaystyle}


\title{Differentiation}

\begin{document}

\maketitle
\begin{enumerate}
\item[Problems:]

$$ $$

%%1%%
\item [1.] Show that if f is differentiable at a then f is
continuous at a.

\

%%2%%
\item [2.] Give an example of a function f which is continuous at a
point a but not differentiable at a.

\

%%3%%
\item [3.] f(x,y) = $\left\{ \begin{array}{ll}
           ax^2+bx+7 & \mbox{if $x<1$} \\  %
            2bx+2a & \mbox{if $x\geq1$} \\  %
\end{array} \right.$

\item [] Suppose that f is differentiable at 1
\item [] Find  a, b.

\

%%4%%
\item [4.] Find $ \frac{\displaystyle dy}{\displaystyle dx} $ for
the following: \
\item [] (1) $\it y=\sin ^3(\tan x)$
\item [] (2) $\it y=\sec(\cos^\frac{1}{2}(x^2+5-\frac{\displaystyle 1}{\displaystyle
x}$))
\item [] (3) $\it y=x^2\cot(\csc x) + x\sin x\cos x$
\item [] (4) $\it y=\sin (x^2 + 1) + \frac{\displaystyle x+2}{\displaystyle x^3+1}$

\

%%5%%
\item [5.] $\it y=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_3x^3+a_2x^2+a_1x+a_0$
\item [] Find $\it y',\enspace y'',\enspace y^{(n)} \enspace and \enspace y^{(n+1)}
$.

\

%%6%%
\item [6.] Find $\frac{\displaystyle dy}{\displaystyle dx}$ \enspace if \enspace $\it
y^3=x^2+\cos{\frac{\displaystyle x}{\displaystyle y^2}}$.

\

%%7%%
\item [7.] Find $\frac{\displaystyle d^2y}{\displaystyle dx^2}$ \enspace if \enspace $\it
xy=\sin x + \cos y $.

\

%%8%%
\item [8.] Let C denote the curve $\it y^4 - 4y^2 = x^4 - 9x^2$.
\item [] (1) Show that (3,2) and (3,-2) are points on the curve
C.
\item [] (2) Find the equation for the tangent to the curve C at
(3,2).
\item [] (3) Find the equation for the normal to the curve C at (3,-2).

\

%%9%%
\item [9.] Let C be the curve described by $$ x=t-\sin t ,\enspace y=1-\cos t $$
\item [] (1) Find $\frac{\displaystyle dy}{\displaystyle dx}
\enspace , \enspace \frac{\displaystyle d^2y}{\displaystyle dx^2}$
\enspace for \enspace t=$\frac{\displaystyle \pi}{\displaystyle 3}$.
\item [] (2) Find the equation for the line tangent to the curve C
at the point defined by t=$\frac{\displaystyle \pi}{\displaystyle
3}$.

\

%%10%%
\item [10.] A, B are walking on the streets that meet at right angles.
A approaches the intersection at 3 m/$\sec$; B moves away from the
intersection at 2 m/$\sec$.
\item [] At what rate is the angle $\theta$ changing and
at what rate is the distance between A and B changing when
 A is 10 m from the intersection and B is 25 m
from the  intersection?


\
  \begin{figure}[!ht]


 \quad\includegraphics[scale=0.75]{Fig1.eps}




\end{figure}


\

%%11%%
\item [11.] (1) $\it f(x)=\sqrt[3]{x}$
\item [] \quad \enspace Find the differential at $\it f $.
\item [] (2) Use the differential to estimate $\sqrt[3]{1000.2}$ .

\end{enumerate}

\end{document}
