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\course \hfill Russell Brown \\
Review 3  \hfill \today

\medskip  For the exam you must know the derivatives of $\sin$,
$\cos$, $\tan$, $\tan^{-1}$, $\sin^{-1}$, $e^x$, $x^a$ and $\ln
x$. More importantly, you should know sum, product, quotient and chain
rules and the techniques of implicit differentiation and logarithmic
differentiation so that you can find derivatives of interesting
functions built up out of the simple functions whose derivatives you
have memorized. 

\begin{enumerate}


\item Differentiate the following functions:
\begin{enumerate}

\item $(x+1) e^x$
\item $(x^2+3x) e^{\sin x}$
\item $ (x^3+ \sin x ) ^2 $
\item $ \cos(e^x)$
\item $ \tan^{-1} ( 1+x^2)$
\item $ e^x \cos x$
\item $ \frac { x^2+ 1} { \sin x }$
\item $  x ^{\sin x}$
\end{enumerate}

\item A ball is thrown in the air. At a time $t$ seconds after it is
thrown, the ball's height is $ h(t) = 30t- 4.9t^2$ meters.  
\begin{enumerate}
\item  What are the units for $ h'(t)$?
\item When is the ball moving up?
\item When does the ball return to the ground?
\item What is the total distance that the ball travels? (Hint: The
answer is not 0.)

\end{enumerate}

\item Use the quotient rule and the derivatives of $\sin $ and $\cos$
to find 
$$
\frac d {dx} \sec x,\  \frac d {dx} \csc x, \  \frac  d {dx} \tan x,\ 
\frac d {dx} \cot x.
$$

\item If $ f'(x)= g(x)$, $g(x)<0$ and $g(x)^2- f(x)^2= 1$, find
$$
\frac d {dx} f^{-1} (x).
$$
Hint: Imitate the procedure for finding $\frac d {dx} \sin^{-1} (x)$.


\item Use implicit differentiation to find 
$$
\frac d {dx} \cot ^{-1} x.
$$
Here $\cot^{-1} $ is the inverse of function $\cot x$ for $x$ in the
interval $(0, \pi)$.

\item Suppose we define $f(x) $ to be the inverse function to $\sin x$
for $x$ satisfying $\pi/2\leq x \leq 3\pi/2$.  
\begin {enumerate} \item Sketch the graph of $f(x)$. Give the domain
and range of $f$.
\item Find the derivative of $f(x)$.  (Hint: This function is not the
function $\sin ^{-1} x$ which is discussed in the text. The derivative
might be different.)
\end {enumerate}


\item  For which $x$ does the tangent line to $y= x^2+1$ pass through
the origin?
\item Find all the tangent lines to the parametric curve
$$ x(t) = \sin(2t), \  y (t) = \cos (t)$$
which pass through $(x,y) = (0,0)$.  (Hint: Graph the curve to find
out how many tangent lines the curve has.)

\item Find all tangent lines at $(x,y) = (0,0)$ for the parametric
curve
$$
x(t) = t^2+t, \  y (t) = t^3-t.
$$


\item Find the tangent line(s) to the curve $x^2+y = y^3$  when $y
=2$. 

\item A function $y= f(x) $ satisfies the equation $ x^2+ y^2 =
2xy$. Find the linear approximation to $f$ at $(x,y) = (1,1)$. 
Use this linear approximation to estimate $f(0.9) $.

\item \begin{enumerate}
\item Find where the curves $ x^2+y^2+1$ and $x^2- 2x + 1 + y^2 = 1$
intersect. (Exact answers please.)

\item Do the curves intersect orthogonally?

   \end{enumerate}

\item Give the  linear approximation of $ f(x) = \sin( \sin (x))$ at
$x= 0$. Use your answer to estimate $ f(0.1)$.

\item Use a linear approximation to estimate $\sqrt {101}$.


\item Evaluate the limits:
\begin{enumerate}
\item  $\lim_{x\rightarrow 0 } \frac { 1 - \cos x } { \sin(x^2)}$
\item  $\lim_{x\rightarrow \infty } {e^{-x} x^2}$
\item $ \lim_{x\rightarrow - \infty } {e^{-x} x^2}$
\item $ \lim _{x\rightarrow 0^+} x^x$
\item $ \lim _{x \rightarrow 2 } \frac {x-2} {x+2}$
\item $\lim _{x\rightarrow 0^+} ( 1 + 2/x)^x$.

\end{enumerate}



\end{enumerate}
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