\magnification 2100
\parindent 0pt
\parskip 10pt
\pageno=1
\hsize 7.7truein
\hoffset -0.5truein
\voffset -0.2truein
\vsize 9.5truein
\def\u{\vskip -10pt}
\def\t{\vskip 10pt}
\def\w{\vskip 1.2cm}
\def\v{\vskip 1.3cm}
\def\s{\vskip -2pt}


Slope of secant line between $(x_1, f(x_1))$ and $(x_2, f(x_2))$ 
\u
\hskip 1in = average rate of change 
\u
\hskip 1in = ${f(x_2) - f(x_1) \over x_2 - x_1}$ 
\u
\hskip 1in = = ${\Delta f(x) \over \Delta x}$ 
\u
where $\Delta x =$ change in $x$ = $x_2 - x_1$ 
\u
and 
$\Delta f(x) = $ change in $f(x)$ = $f(x_2) - f(x_1)$

Slope of tangent line to $f$ at $x_1$ = instantaneous rate of change $$= lim_{x_2 \rightarrow x_1} {f(x_2) - f(x_1) \over x_2 - x_1}$$ $$= lim_{x_1 + h \rightarrow x_1} {f(x_1 + h) - f(x_1) \over x_1 + h - x_1}$$ $$ = lim_{h \rightarrow 0} {f(x_1 + h) - f(x_1) \over h}$$  



 Definition  $f'(a) = $ slope of tangent line to $f$ at $a$

\rightline{ = $lim_{h \rightarrow 0} {f(a + h) - f(a) \over 
h}$.}

If $f(x) = 2x - 4$, then $f'(8) = $

If $g(x) = 3$, then $g'(1) = $

If $h(x) = |x|$, then $h'(5) = $  

\hskip .8in and $h'(-5) = $

\eject

 Definition:  Given $f$, then define the function $f'$ (the derivative of $f$) as follows: 
\vskip -5pt
$f'(x) = $ slope of tangent line to $f$ at $x$ = $lim_{h \rightarrow 0} {f(x + h) - f(x) \over h}$.  

$x$ is in the domain of $f'$ if $x$ is in the domain of $f$ and the above limit exists.

If $f(x) = 2x - 4$, then $f'(x) = $

If $g(x) = 3$, then $g'(x) = $

If $h(x) = |x|$, then $h'(x) = $  

\t
Suppose $f(x) = -2x + 12$ represents the distance traveled from home in 
miles after $x$ 
hours.  Find the average velocity between $x = 1$ and $x = 3$.
What are the units?

\w
\t\t

Find the instantaneous velocity at $x = 1$:
What are the units?
\s\s
\w

Find the acceleration at $x = 1$:
What are the units?
\eject

Suppose $f(x) = -2x + 12$ represents the population of a household
$x$ years after 2000.
Find the average change in the population
between 2001 and 2003 (i.e., $x = 1$ and $x = 3$).
What are the units?

\v
Find the instantaneous change in the population at $x = 1$:
What are the units?

\v
Suppose $f(x) = 8$ represents the distance traveled from home in 
miles after $x$ 
hours.  Find the average velocity between $x = 1$ and $x = 3$.
What are the units?

\v

Find the instantaneous velocity at $x = 1$:
What are the units?

\v

Find the acceleration at $x = 1$:
What are the units?

\eject

Suppose $f(x) = 8$ represents the population of a household
$x$ years after 2000.
Find the average change in the population
between 2001 and 2003 (i.e., $x = 1$ and $x = 3$).
What are the units?



\v
Find the instantaneous change in the population at $x = 1$:
What are the units?

\v


Suppose $f(x) =  {x + 3 \over 4x + 1} + 2$ represents the distance 
traveled 
from home in 
miles after $x$ 
hours.  Find the average velocity between $x = 1$ and $x = 3$.
What are the units?

\v\t\t\t\t

Find the instantaneous velocity at $x = 1$:
What are the units?

\v

\eject

Suppose $f(x) = {x + 3 \over 4x + 1} + 2$ represents the population of a 
household
$x$ years after 2000.
Find the average change in the population
between 2001 and 2003 (i.e., $x = 1$ and $x = 3$).
What are the units?



\v
Find the instantaneous change in population at $x = 1$:
What are the units?

\v

The volume of a right circular cylinder is given by the formula 
$V = \pi r h^2$ where $r$ is the radius of the base in meters and $h$ is 
the height 
of the cylinder in meters.  

Find the rate of change of the volume of the cylinder with respect to the 
height?


\vfill
\vfill


Find the rate of change of the volume of the cylinder with respect to the 
radius?





\end