The applet is a function graph-plotter that can find derivatives and locate critical points on up to five simultaneous graphs..

Functions can be entered in the input field or retrieved from the “history list”. Add to the list the derivative of a selected function, or zoom in and solve for roots, local max/min, points of inflection, and intersect-ions of curves.

Using the function input field: At the top left of the screen is an input field for adding new functions to the displayed list of functions. Remember that all such inputs must be ended with the Enter Key. For example type x^x (and hit Enter). Type sin(x^2). Type x^3-3*x+1. The syntax for expressions follows the basic rules of algebra. The list of functions provided initially gives examples. And syntax errors are pointed out at the time you make them. As an example, type sin(x in the input field (i.e. omit the closing parenthesis). Note that a suggestion is made for correcting the error, which you do by editing the input field (type the closing parenthesis).

Setting the number of “active” functions: Below the function list are several buttons and a pull down menu. Initially the menu choice is “1”, meaning that only one of the function in the list is active. Select another number from the menu to make more functions active. Note that active functions are displayed with a colored box to their left, the color providing the key to the graph of the function in the plot window. The check mark in the colored box can be toggled by clicking the box, and this results in showing or hiding the graph. Up to five functions can be active.

Selecting a function in the list, and operating on the selected function: Clicking on a function in the list will select it (hiliting it with yellow). When selected, the function can be operated upon with the three buttons below the button list. The DELETE button erases the selected function from the list. The RECALL button moves an inactive function to the top of the list where it becomes active and its graph is displayed. The d / dx button adds the derivative of the selected function to the top of the list. The derivative is donoted by an expression of the form
D ( f(x) ). When the derivative is hilited it displays a symbolic expression for the derivative function. Try, for example, selecting the function e^sin(x) and pressing RECALL. It moves to the top of the list and its graph is displayed. Select it again and press d / dx. Its derivative now appears at the top of the list. Select the derivative (noting that the algebraic expression for the derivative is displayed) and press d / dx again. The second derivative is now displayed at the top of the list. If you have set the number of active functions to 3 you will see the graphs of e^sin(x), D(e^sin(x)), and DD(e^sin(x)) displayed in the plot window. Note on entering derivatives: Do not type an expression such as “D(sin(x))” into the function input field. It will be interpreted as “D*sin(x)”, not as the derivative of sin(x). Use the d / dx button to enter a derivative.

ZOOM, SOLVE, SELECT, and LABLE modes: Below the plot window are four buttons.. Each of these buttons selects a particular operating mode of the applet and displays additional controls appropriate to the mode chosen. By default ZOOM is selected and additional buttons for zooming in or out are provided. Experiment with zooming in by dragging a rectangle around a part of the graph of particular interest. Repeated zooming in can be useful for magnifying a function in the vicinity of a root or a limiting value. The Reset Zoom button restores the default coordinate system. And the ZoomX and ZoomY buttons are convenient for zooming out. Often one wants to change the scale for the x or y coordinates independently. As an example type x^8 + 45*x^5 - 367*x^2 + 1000*pi in the input field (and hit Enter). Set the number of active functions to 1. Note that no graph appears. Now click on ZoomY several times to change the scale of the plot, bringing the graph of the function into view. Note that you can now see that the function has a root near x = 2. Zoom in on this root by dragging a small rectangle around it. Do this several times to refine your knowledge of the root. Finally, clicking Reset Zoom restores the default coordinates. The SOLVE, SELECT, and LABEL buttons are discussed below.

Using SOLVE mode: Click on the SOLVE button to initiate SOLVE mode. In this mode you carry out most actions by dragging rectangles with the mouse. The strategy is to drag a rectangle around a point of interest on a curve, for example around a zero, a relative maximum, or an inflection point. If your rectangle has only one function passing through it, then the applet attempts to find and identify the “interesting point” that you chose. Often it is useful to first zoom in to an area of the plot containing the point of interest, especially if there are several graphs showing and it would otherwise be difficult to select only one function. A single mouse click in SOLVE mode will zoom in on the point you clicked, coordinates changing by a factor of two. Repeating this several times allows the point of interest to be clearly visible, after which you can drag a rectangle around it without fear of including neighboring curves. A Reset Zoom button is provided for convenience. If you drag a rectangle around a point of intersection of two curves the applet attempts to find and identify the point of intersection. If your rectangle has more than two curves passing through it, the operation is ignored. In some cases when curves are very close together you may want to first zoom in on the intersection point. Alternatively, you can “deactivate” the curve in which you have no interest (and that is in the way) by clicking its color box in the function list. This temporarily removes its graph and lets you free to drag a rectangle around the intersection point of the curves of interest.

Using SELECT mode: Click on the SELECT button to initiate SELECT mode. In this mode you can select one of the displayed functions by clicking on it. The curve is hilited, and the function is also hilited in the function list. When a function is selected, any of the DELETE, RECALL, or d / dx actions can be performed on it. Selecting a function in this way is equivalent to clicking on its name in the function list. When the graphs of many functions are displayed it is sometimes convenient to select one in order to identify to which expression in the function list it corresponds.

Using LABEL mode: Click on the LABLE button to initiate LABEL mode. In this mode you can add a label to a curve by clicking on the curve. Move the label by clicking the curve in a different spot. Label several of the curves to easily identify them in the display. Remove all labels by clicking on the Clear labels button. (Note: when a curve is very steep you may need to click several times to effect the labeling. Try clicking near a point where the curve shows a one-pixel break.)

Experiment with the applet: Add functions of your choice. Delete ones in which you have no interest. Set the number of active functions to a suitable value for the problem you want to solve. Experiment with zooming, using it to get information about roots of functions or other desired information. Experiment with SOLVE mode. In general the tools provided by the applet, used in concert, give considerable power to investigate the behavior of functions of your choice.

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