HANAH STRIKES AGAIN!

To calculate the slope of a line tangent to $f$ at $x = a$, most graphing calculators use Hanah’s method from problem 3-35, formally called the symmetric difference quotient, shown below.

$\lim\limits _ { h \rightarrow 0 } \frac { f ( x + h ) - f ( x - h ) } { 2 h }$

1. Use the symmetric difference quotient to determine $f ^\prime$ if $f(x) = 3x − 2$.

   Step (a):

   $f'(x)=\lim\limits_{h\rightarrow 0}\frac{(3(x+h)-2)-(3(x-h)-2)}{2h}$

   More Steps (a):

   Expand the numerator.
   Combine like terms.
   Factor out an $h$.
   Cancel out the $h$.
   Evaluate the limit $h\rightarrow 0$.... this is $f ^\prime (x)$.

2. Use this symmetric difference in your graphing calculator to graph $f ^\prime$ for $f(x)=\sin(x)$ for $h=0.001$.

   Steps (b):

   $f'(x)\approx \lim\limits_{h\rightarrow 0.001}\frac{(3(x+0.001)-2)-(3(x-0.001)-2)}{2(0.001)}=\underline{ \ \ \ \ \ \ \ \ \ \ \ }$

   Graph that on your calculator. Also sketch the derivative function you found in part (a). The sketches should look a lot alike.