You can use GeoGebra
to animate an approximation of instantaneous rate of change.
GeoGebra
Instruction 1
Algebra View
and Graphics View
under View
in Menu
. f
into Algebra View
: f = $\dots $
I recommend that you use a quadratic function, so that the approximation becomes as clear as possible.
h
into the next row in Algebra View
and press Enter
. This input makes GeoGebra
prepare a slider for you. Right-click the row with the slider and click Gear (Settings)
. Click the Slider
tab, and set
Min
to $-1$
Max
to $1$
Speed
to $0.05$
Algebra View
as follows: s = <the x-coordinate of your point>
Now you get another slider.
Tangent(<Point>, <Function>)
and replace <Point>
with s
(you only need the $x$-coordinate) and <Function>
with f
. The slope of the tangent is the instantaneous rate of change at the point $(s,f(s))$.
Line(<Point>, <Point>)
where
The first <Point>
field is replaced with (s,f(s))
The second <Point>
field is replaced with (s+h,f(s+h))
This line is the approximation of the tangent you drew in the previous step. That means that the slope of this line is an approximation of the slope of the tangent.
Now, you can adjust the value of h
using the slider. Notice how the line you drew looks more and more like the tangent when h
tends to 0. You can also adjust the number s
using the slider to move the tangent.