                   # Derivative as the Slope of the Tangent

Introduction - Features - How to Use the Applet

# 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 y-values. 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 y-range" is checked (the default), then both the x and y-range values are used. If it is unchecked, the applet will only use the x-range values and compute the corresponding range for the y-values.

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 x-coordinate of the point where the tangent will be drawn. Once these are specified, the corresponding y-coordinate 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 x-axis. 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 x-axis. 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:

1. Enter any function.

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

3. 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.

4. 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:

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

2. 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.

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

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