\begin{homework}\
\label{homework:3}

\medskip

\exercise %1
Draw the graph of the function $y=f(x)=\sqrt{169-x^2}$ between $x=0$
and $x=13$.  Find the slope $\Delta y/\Delta x$ of the chord between the
points of the circle lying over (a) $x=12$ and $x=13$, (b) $x=12$ and
$x=12.1$,  (c) $x=12$ and $x=12.01$, (d) $x=12$ and $x=12.001$.  Now use
the geometry of tangent lines on a circle to find (e) the exact value of the
derivative $f'(12)$.  Your answers to (a)--(d) should be getting closer and
closer to your answer to (e).

\smallskip

\exercise %2.
Use geometry to find the derivative $f'(x)$ of the function
$f(x)=\sqrt{625-x^2}$ in the text for each of the following $x$: (a) 20,
(b) 24, (c) $-7$, (d) $-15$.  Draw a graph of the upper semicircle, and
draw the tangent line at each of these four points.

\smallskip

\exercise %3.
Let $y=f(t)=t^2$, where $t$ is the time in seconds and $y$ is the distance
in meters that an object falls on a certain airless planet.  Draw a graph
of this function between $t=0$ and $t=3$.  Make a table of the average
velocity of the falling object between (a) 2 sec and 3 sec, (b) 2 sec and
2.1 sec, (c) 2 sec and 2.01 sec, (d) 2 sec and 2.001 sec.  Then use algebra
to find a simple formula for the average velocity between time $2$ and time
$2+
\Delta t$.  (If you substitute $\Delta t=1,\>0.1,\>0.01,\>0.001$ in this
formula you should again get the answers to parts (a)--(d).)  Next, in your
formula for \index{ average velocity} average velocity (which should be in simplified form) determine
what happens as $\Delta t$ approaches zero.  This is the instantaneous
velocity.  Finally, in your graph of $y=t^2$ draw the straight line
through the point $(2,4)$ whose slope is the \index{ instantaneous velocity} instantaneous velocity you
just computed.

\smallskip

\exercise %4.
If an object is dropped from an 80-meter high window, its height $y$ above
the ground at time $t$ sec is given by the formula $y=f(t)=80-4.9t^2$.
(Here we are neglecting air resistance; the graph of this function was
shown at the beginning of the first section.)  Find the average velocity of
the falling object between (a) 1 sec and 1.1 sec, (b) 1 sec and 1.01 sec,
(c) 1 sec and 1.001 sec.  Now use algebra to find a simple formula for the
average velocity of the falling object between 1 sec and $1+\Delta t$ sec.
Determine what happens to this average velocity as $\Delta t$ approaches 0.
That is the instantaneous velocity at time $t=1$ sec (it'll be negative,
because the object is falling).

\smallskip

\exercise %5
Draw the graph of the function $y=f(x)=1/x$ between $x=1/2$ and $x=4$.
Find the slope of the chord between (a) $x=3$ and $x=3.1$, (b) $x=3$ and
$x=3.01$, (c) $x=3$ and $x=3.001$.  Now use algebra to find a simple
formula for the slope of the chord between $(3,f(3))$ and $(3+\Delta
x,f(3+\Delta x))$.  Determine what happens when $\Delta x$ approaches 0.
In your graph of $y=1/x$, draw the straight line through the point
$(3,1/3)$ whose slope is this limiting value of the difference quotient as
$\Delta x$ approaches 0.

\smallskip

\exercise %6.
Find an algebraic expression for the difference quotient $\bigl(f(1+\Delta
x)-f(1)\bigr)/\Delta x$ when $f(x)=x^2-(1/x)$.  Simplify the expression as
much as possible.  Then determine what happens as $\Delta x$ approaches 0.
That value is $f'(1)$.

\smallskip

\exercise %7.
Draw the graph of $y=f(x)=x^3$ between $x=0$ and $x=1.5$.  Find the slope
of the chord between (a) $x=1$ and $x=1.1$, (b) $x=1$ and $x=1.001$, (c)
$x=1$ and $x=1.00001$.  Then use algebra to find a simple formula for the
slope of the chord between $1$ and $1+\Delta x$.  (Use the expansion
$(A+B)^3=A^3+3A^2B+3AB^2+B^3$.)  Determine the limit as $\Delta x$
approaches 0, and in your graph of $y=x^3$ draw the straight line through
the point $(1,1)$ whose slope is equal to the limit you just found.


\end{homework}

