The function and derivative.
Here is a quick and easy way to visualize how a function and its derivate relate. Oftentimes, one
of the most frustrating things for Calculus students is visualizing how the derivative of a function
relates to the function itself. This GeoGebra construction will hopefully help you out as a good
supplement.
Step 1
First step is to create your grid. Most of you already know how to do this, but I do want to make
you aware of a few options that are pretty helpful. Right-click on either axis and you will notice
the Preferences slide from the right.
In the Preferences pane, under the Basic tab, make sure Bold is checked. This option will make
it easier to see the grid.
Step 2
Next, select the xAxis tab and check the option labeled Distance. This will give you access to a
pull-down menu, select the option "π/2". Your x-axis will now change to display radian values.
The function and derivative.
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Step 3
Now, let's begin with the construction. You can choose any function you like, but try to keep in
mind that you may want a graph of a function that displays key points (i.e., inflections, roots, min/
max).
Simply type your function in the Input bar toward the bottom of the GeoGebra window.
For this example, I have chosen a function that seems to work pretty well for this purpose:
f(x) = 2sin(x) + 6cos(x)sin(x)
You may also want to change some of the function’s graph display properties such as its width
and color. In this case, I have made the line red and changed its size width. You can do this by
right-clicking on the function itself, selecting "Object Properties" and changing the properties
under the Color and Style tabs.
The function and derivative.
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Step 4
Create a point on the graph of the function using the Point tool. You may want to change its style
settings (in my case I changed its size and color to make it easier to see).
The function and derivative.
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Step 5
Next, create a tangent line through the point (in my case, Point A). When creating a tangent line
through the point, be sure to select the point AND the graph of the function.
Step 6
You want to record the slope of the tangent line. To do this, simply select the "Slope" option from
the Toolbar. You will need this number for our next and final step. You can hide the visual of the
slope as you see below, this may create less clutter on your graph. To do this, simply right-click
on the "Number" on the left-hand window and deselect "Show Object".
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Step 7
Now create a point that lies on the graph of the derivative of the function. This point will change
dynamically as you move your point along the graph of the original function. In my example,
Point B is on the graph of f'(x), as I move Point A, Point B moves along the graph of the
derivative.
To do this, type into the Input bar: (x(A),m)
This will create a point on the graph of the derivative of the function. You will want to enable the
trace feature for Point B. To do this, right-click on Point B and select "Trace On".
Note that your labels may be different than the ones in this example so be sure to use the labels
in your construction!
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Step 8
To help students visualize this more clearly, you will want to show the graph of the derivative of
the function.
Simply type into the Input bar: derivative[f(x)]
This command will graph the derivative of your function. In this example, it is shown as "g(x)".
You may want to change the style settings on this graph.
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And there you have it! You can add some optional functionality to the file. For example, if you are
giving this file to students, you might want to add some sliders that students can move to
manipulate the point of the function. Or, turn the trace feature off and hide the point on the
derivative to have students guess what the derivative will look like at different intervals of the
original function.
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