Introduction
This applet
illustrates the fact that the derivative of a function y = f(x) at
x = a, denoted f'(a), is the slope of the tangent to the graph of
y = f(x) a x = a. The user can plot a function, and select points
on the graph. As the points are selected, the tangent to the graph
at the chosen point is optionally drawn, its slope is displayed and
plotted as a new point. If the user selects enough points, the graph
of the derivative will become evident.
Features
There are
six areas on the applet the user can interact with.
Viewing
Window Parameters.
In this
area, the user specifies the range for the x and yvalues. The default
is that x and y are between 10 and 10. Any number as well as any
expression resulting in a number can be used here. For example, when
trigonometric functions are studied, the user may prefer to use multiples
of pi such as 2*pi or 4*pi.
If "use yrange"
is checked (the default), then both the x and yrange values are used.
If it is unchecked, the applet will only use the xrange values and
compute the corresponding range for the yvalues.
The RESET ZOOM
button is used to reset the x and y ranges to their default values.
Function
information
In this
area, the user specifies the function to use and the xcoordinate
of the point where the tangent will be drawn. Once these are specified,
the corresponding ycoordinate of the point will be displayed. If
the check box "Show Tangent" is checked, the tangent will
also be displayed.
Derivative
Information
Use the
LEFT and RIGHT buttons or arrow keys which are to move the point where
the tangent is being drawn. As the point moves, the various related
fields are updated, the slope is being plotted for every new point.
In other words, if the point selected has coordinates (a, f(a)), we
also plot the point (a, slope) where slope is the slope of the tangent
at x = a. Since the slope is the derivative f'(a), what we are really
plotting, one point at a time, is the derivative of f. For example,
the user can experiment with this. If the point selected on the graph
corresponds to a local maximum or minimum, the slope of the tangent
will be zero, so the point corresponding to the slope being plotted
will be on the xaxis. If the point selected is on a portion where
the graph is rising, the slope is positive, so the point corresponding
to the slope being plotted will be above the xaxis. And so on.
By default, the
point moves one pixel at a time. However, this value can be changed
by using the choice menu entitled "increment (pixels)".
The point will then move by whatever number of pixels is selected.
Keep in mind that the value one pixel represents depends on the zoom
level.
Control
Buttons
The PLOT
button is used to plot the function after it has been entered.
The CLEAR ALL
button erases all the areas. The applet will look as if you had just
started it.
Plot
Area
This areas
displays the graph, the various points chosen, the tangent line and
the slope . The only interaction with this area is via the mouse.
The user can click near the graph to select points defining the tangent.
The user can also draw a virtual box by clicking on one of its corner
and dragging the mouse to the other corner. The graph will be zoomed
into this box, the fields in the Viewing Window Parameters area will
be updated accordingly.
Message
Area
There is
no interaction with this area. It is used to display hints on what
the user should do next as well as error messages if the wrong operation
is performed. Always look at what is displayed in this area.
User
Actions
The user can
do the following:

Enter any
function.

Pick the
point (a, f(a)) where the tangent will be drawn by either clicking
on the graph, or specifying the value of a.

Pick more
points until the graph of the derivative becomes evident by repeating
step 2 or by using the LEFT or RIGHT buttons or arrow keys in
the Derivative Information area.

Zoom in an
area by clicking in one corner of a virtual box, and dragging
the mouse to the opposite corner of that box. This allows the
user better control on the increment by which the point on the
graph moves if the point is moved using the left and right buttons.
The point will always move by one pixel (default) or by whatever
value has been selected in the "increment" choice item.
Keep in mind that the value one pixel represents depends on the
zoom level.
How
to Use the Applet
Use the applet
as follows:

If necessary,
change the values in the Viewing Window Parameters area, though
the default values should be fine in many cases.

Enter a function.
Nothing can happen unless a function has been entered. Use the
same syntax as in the PlotSolve or the Java Math Pad applets.

Next, specify
the point where the tangent will be drawn.

Specify more
points, until the graph of the derivative becomes evident.