\magnification 2100
\parindent 0pt
\parskip 10pt
\pageno=1
\hsize 7.2truein
\hoffset -0.2truein
\vsize 9.5truein
\def\u{\vskip -10pt}
\def\t{\vskip 10pt}
\def\w{\vskip 1.2cm}
\def\v{\vskip 1.3cm}


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

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) = $


 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?

\w

Suppose $f(x) = -2x + 12$ represents the cost of stock after  
$x$
days from purchase.  Find the average change in the cost of the stock 
between $x = 1$ and $x = 3$.
What are the units?

\v
Find the instantaneous change in the cost of the stock 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

Suppose $f(x) = 8$ represents the cost of stock after  
$x$
days from purchase.  Find the average change in the cost of the stock 
between $x = 1$ and $x = 3$.
What are the units?

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

\v


Suppose $f(x) =  {x + 3 \over 4x - 5}$ 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

Suppose $f(x) = {x + 3 \over 4x - 5}$ represents the cost of stock after  
$x$
days from purchase.  Find the average change in the cost of the stock 
between $x = 1$ and $x = 3$.
What are the units?

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

\v

\vfill

HW \hfil \break 
2.7) 1, 2, 3, 9, 13, 15, 17, 27\hfil \break
2.8) 3, 4, 5, 7, 15, 25, 29, 33, 35, 36\hfil \break
2.9) 4, 7, 9, 21, 27, 29, 37, 38\hfil \break
3.1) 3, 23, 29, 31, 45 \hfil \break
and

 





\end