                   # Introduction

The slope of the tangent to the graph of y = f(x) at x = a can be approximated as follows:

1. Consider the secant line through the points (a, f(a)) and (a+h, f(a+h)) for a certain value of h.

2. Its slope is .

3. When h approaches 0, the point (a+h, f(a+h)) gets closer to the point (a, f(a)). The secant line becomes the tangent to the graph of y = f(x0 at x = a. Therefore, the slope of the secant approaches the slope of the tangent.

This applet illustrates this concept. It allows the user to enter a function, to select the points (a, f(a)) and (a+h, f(a+h)), to change h so that the point (a+h, f(a+h)) gets closer to the point (a, f(a)).

# 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.

## Tangent line 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 as well as the slope of the tangent will be displayed.

## Secant Line Information

In this area, the user specifies the value of h for the second point defining the secant. Once this is done, the coordinates a+h and f(a+h) will be displayed as well as the slope of the secant. The user can also use the left and right buttons to move this second point as well as the left and right arrow keys. As this happens, all the fields in this area will be updated accordingly and the secant line will be redrawn.

## 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 and secant lines. The only interaction with this area is via the mouse. The user can click near the graph to select the first point defining the tangent, and the second point defining the secant. 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 the second point (a+h, f(a+h)) by either clicking on the graph or specifying the value of h.

4. Change the point (a+h, f(a+h)) by either clicking in another location, specifying a new value for h, or using the buttons or the arrow keys to move left or right . If h is chosen to be smaller and smaller, the user will be able to observe that the secant line becomes the tangent line and that the slope of the secant approaches the slope of the tangent.

5. 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 in h, if the second point defining the secant is moved using the left and right buttons. The point will always move by one pixel. However, 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 the second point for the secant line.

5. Change the second point defining the secant line. In doing so, the user can verify that as h approaches 0, the secant becomes the tangent. This also allows the user to evaluate the expression as h approaches 0  