How to Animate Instantaneous Rate of Change in GeoGebra

1.

Open Algebra View and Graphics View under  View in  Menu.

2.

Type a function f into Algebra View:

f = $\dots$

I recommend that you use a quadratic function, so that the approximation becomes as clear as possible.

3.

Type 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 $0.05$

4.

Select a point on your graph, ideally not one that is an extremum. Type the $x$-coordinate of your point in Algebra View as follows:

s = <the x-coordinate of your point>

Now you get another slider.

5.

Use the command

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 $(s,f(s))$.

6.

Use the command

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.