\def\wsn{12}
\input worksheet.tex

\item{1.}  Imagine a road on which the speed limit is specified at
every single point.  In other words, there is a certain function $L$
such that the speed limit $x$ miles from the beginning of the road is
$L(x)$.  Two cars A and B, are driving along this road;  car A's
position at time $t$ is $a(t)$, and car B's is $b(t)$.

\medskip
\itemitem{a)}  What equation expresses the fact that the car A always
travels at the speed limit?  
\itemitem{}(Hint: the answer is {\it not} $a'(t)=L(t)$.)

\medskip
\itemitem{b)}  Suppose that A always goes at the speed limit, and that
B's position at time $t$ is A's position at time $t-1$.  Show that B
is also going at the speed limit at all times.

\medskip
\itemitem{c)}  Suppose B always stays at a constant distance behind A.
Under what circumstances will B still always travel at the speed limit?

\bigskip
\item{2.}  Prove that it is impossible to find two differentiable
functions $f(x)$ and $g(x)$ for which $f(0)=g(0)=0$ and which satisfy
$f(x)g(x)=x$ for all $x$.  
\item{}  (Hint: differentiate.)

\bigskip
\item{3.}  Assume that $f(x)$ is a differentiable function and that
the values of $f(x)$ and its derivative at the points $x=0,1,2$, and
$3$ are given by:
$$
{\eqalign{f(0)&=3\cr f'(0)&=-1\cr}}\qquad
{\eqalign{f(1)&=5\cr f'(1)&=0\cr}}\qquad
{\eqalign{f(2)&=-2\cr f'(2)&=3\cr}}\qquad
{\eqalign{f(3)&=6\cr f'(3)&=1.\cr}}
$$
Let $g(x)=x^2-3x+2$.  For each function below, calculate the
derivative at the given point.
$$
\matrix{
\pt a f(x)+g(x);\quad x=0\hfill&&
\pt b \displaystyle{f(x)\over g(x)};\quad x=1\hfill&&
\pt c f(x)g(x);\quad x=2\hfill&&
\pt d \displaystyle{f(x)g(x)\over f(x)+g(x)};\quad x=3\hfill\cr\cr
\pt e f(g(x));\quad x=0\hfill&&
\pt f f(g(x));\quad x=1\hfill&&
\pt g g(f(x));\quad x=2\hfill&&
\pt h g(f(x));\quad x=3\hfill\cr}
$$

\bigskip
\itemitem{4.\hskip12pt a)}  Prove that the formula for the derivative
for an inverse function is
$$
(f^{-1})'(x)={1\over f'(f^{-1}(x))}.
$$
(Hint:  Let $g(x)=f^{-1}(x)$, then $f(g(x))=x$.  Differentiate.)

\medskip
\itemitem{b)}  Find $f^{-1}(x)$ given that $\displaystyle
f(x)={2x-3\over x+2}$. 

\medskip
\itemitem{c)}  Differentiate $f^{-1}(x)$ from part b) and compare with
the derivative you get by applying the formula in part a).

\bigskip
\item{5.}  Let $f(x)=x^3+3x^2+3x+5$.  Does $f$ have an inverse?  How
do you know?  If $f$ has an inverse, determine $f^{-1}(5)$,
$(f^{-1})'(5)$, $f^{-1}(12)$, and $(f^{-1})'(12)$.

\eject
\item{6.}  When you see the function $x\mapsto\sin x$ it is
assummed that $x$ is given in radians.  Define a {\bf different}
function $f$ to be the function such that $f(x)$ is the value of the sine
of an angle which measures $x$ {\bf degrees}.  (NOTE: These are indeed
different functions!  For most values of $x$, $f(x)\neq\sin x$.  Why
not?)  Let $g$ be defined similarly for cosine.

\medskip
\itemitem{a)}  Express $f$ and $g$ in terms of sin and cos.

\medskip
\itemitem{b)}  What is $\displaystyle{df\over dx}$?  What is
$\displaystyle{dg\over dx}$?  (Hint: Use part a) and the chain rule.)

\medskip
\itemitem{c)}  Express $\displaystyle{df\over dx}$ and $\displaystyle
{dg\over dx}$ in terms of $f(x)$ and $g(x)$.  (No mention of sin or cos
allowed.)

\medskip
\itemitem{d)}  Is it still true that $(f(x))^2+(g(x))^2=1$?

\medskip
\itemitem{e)}  Why don't we use the unit of degrees in calculus?

\bigskip
\item{7.}  Find $\displaystyle{dy\over dt}$ given that $\displaystyle
{y={3x^3\over2x^2+1}}$ and that $x=\tan(t^2)$.  

\bigskip
\item{8.}  For this problem, you may want to review your result for
the derivative of an inverse from Problem 4.

\medskip
\itemitem{a)} What is $(\arccos(x))'$?  Write an
expression which is free of trig functions.
\itemitem{}  (Hint:  Use the formula in part $a$.  To help you simplify,
draw a triangle.)

\medskip
\itemitem{b)}  Let $y=\tan^2(\arccos(x))$.  Find $\displaystyle
{dy\over dx}$ using the chain rule.

\medskip
\itemitem{c)}  Find another expression for $y$ which doesn't mention
trig functions. 
\itemitem{}  (Hint:  Again, draw a triangle.)


\medskip
\itemitem{d)}  Find $\displaystyle{dy\over dx}$ from your expression in
part d).  How does it compare with your answer in part a)?
  



\bye