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HW Supplement: Equation of Tangent Line Namt Let

`f(x)=0.1x^(3)-2.5x+3.4`

a). Use a graphing tool to graph of

`f(x)`

for the window

`x[-6,6]`

and

`y[-5,10]`

and find

`f(0),f(6)`

, and

`f(4)`

. Then plot and label the three points and reproduce a sketch of the graph below. b. Use a ruler to sketch, above, the secant line between the points where

`x=0`

and

`x=6`

. The slope of this secant line is the AVERAGE rate of change of

`f(x)`

over the interval

`0,6`

. c. Find the average rate of change of

`f(x)`

over the interval

`0,6`

. (show work). d. Use a ruler to sketch, above, the tangent line to

`f(x)`

at the point where

`x=4`

. e. Is the slope of the tangent line greater or less than the slope of the secant line drawn? (guess) f. Use the basic rules to find

`f^(')(x)`

.

`f(x)=0.1x^(3)-2.5x+3.4,f^(')(x)=`

The instantaneous rate of change of

`f`

at

`x=4`

is the slope of the tangent line at that point. The slope of the tangent to a function at a point is the derivative at that point. g. Find

`f^(')(4)=`

and compare this slope to the slope of the secant from (c). h. Use your results in part

`g`

to find the equation of the tangent line to

`f(x)`

at

`x=4`

. Pt:

`(4,f(4))`

Slope:

`m=f^(')(4)`

Equation of line:

`y-y_(o)=m(x-x_(o))`