(Version of 26 October 1999)
The xFunctions applet is meant as an educational tool for exploring several topics in calculus and pre-calculus mathematics. It can also be used to have fun playing around with mathematical functions. The applet includes most of the features of my popular xFunctions program for Macintosh, which is still available for downloading from my home page, although it is getting a bit out of date. xFunctions is designed to be easy to use. It can, for example, be used in a course with very little class time required to explain how to use it. In addition to basic graphing, xFunctions lets you do animated graphs, parametric curves, derivatives, Riemann sums, integral curves, and 3D graphs.
Instructions and more info can be found later on this page!
There are two versions of the applet. In one version, xFunctions appears on the Web page. In the other version, the only thing that appears on the Web page is a button. Clicking on the button will open xFunctions in a separate window. This has the advantage that you can resize the window. You'll see both versions below. It is also possible to run xFunctions as a standalone application from the command line (if you download it).
You can add your own examples to xFunctions, which can be accessed from a menu in the applet. The examples can be read from a file or from applet parameters. (Unfortunately, I haven't provided a very convenient way to create the examples.) On this page, only the "launcher" version of the applet uses such examples, but they can be used with both versions and they can be used with xFunctions when it is run as a standalone application.
Instructions on using examples can be found on another page.
Although it is not public domain, xFunctions can be freely used and distributed for non-commercial purposes. (If you find a commercial purpose for it, please ask my permission before using it.) You can download a copy of the xFunctions Web site -- the applet code, two Web pages, a README file, and as sample file of examples. The files are available in several different formats at the following links:
I do not plan to develop xFunctions any further, except to fix bugs. However, a lot of the design philosophy that went into it will survive in my brand new project, "MathBeans," which you can read about at my MathBeans Web site.
(By the way, the "x" in xFunctions stands for "Eck's".)
Department of Mathematics and Computer Science
Hobart and William Smith Colleges
Geneva, New York, 14456, USA
Note: This applet is a 250 kilobyte download.
If you are on a slow connection, please be patient.
(Why not read the instructions, below, while you wait?)
Launcher Version, with added examples
(Press the button to open a separate, resizable window.)
When the applet starts up, it is showing a "Main Screen" where you can see the graph of functions. The available functions are shown in a list on the left. Click on a function to view its graph. You can define your own functions to add to this list using the three buttons at the lower left of the Main Screen. There are three ways to define new functions: using expressions (such as 0.5*x^2+sin(3*x-2)), by giving the graph of the function, or by listing a table of (x,y) pairs. There is a separate input screen for each of these input methods. To get back to the Main Screen from one of these input screens, you have to press a "Done" or "Cancel" button. The pop-up menu at the top of the applet can be used to go to seven other screens. Each screen is a separate "utility" that allows you to play with functions in a different way. When you enter functions into the utility screens, you can use any new functions that you have defined, as well as the built-in functions.
In the "Launcher Version," you'll find a few extra functions in the list on the Main Screen. You'll also find some extra entries in the pop-up menu. These extra entries are examples that will take to one of the utility screens, and load an example in to that screen. (Information about making examples for xFunctions can be found here -- but to do that, you need to make your own Web pages.)
Now, just go ahead and play!
More Detailed Instructions
First some general ideas
- Expressions in xFunctions can include the operations +, -, *, /, and ^, where ^ represents exponentiation (this is, x^3 is x*x*x). You always have to type multiplication explicitly as *. When you use a function such as sin or f in an expression, you have to put parentheses around its argument. That is, you have to say "sin(x^2)" rather than "sin x^2". You can use the constants e and pi in expressions.
- Built-in functions are shown on the main screen. Some might not be obvious: "abs" is absolute value. "sqrt" is square root and cubert is the cube root. "ln" is the natural logarithm and "exp" is exponentiation in the base e (same as e^x). "round(x)" is the nearest integer to x, "floor(x)" is the integer just below or equal to x, "ceiling(x)" is the integer just above or equal to x, and "trunc(x)" in the integral part of x after any decimal part is thrown away.
- Graph limits and other limits can be set by filling in the text boxes labeled xmin, tmax, and so on. Usually, if you press return in any text box, the graph will be redrawn, so you don't have to use the buttons too much. Many screens have buttons for zooming in and out on the graph. A "Restore" button will restore the original limits on the graph. An "Equalize" button will reset the xy limits on the graph so that the scales on the horizontal and vertical axes are equal. (This makes circles look like circles instead of ellipses, and it lets you estimate slope visually.)
- The mouse can be used for zooming on some of the applet's screens. Click with the right mouse button at a point (or command-click on Macintosh) to zoom in on that point. Click with the middle mouse button (or option-click or ALT-click) to zoom out from the point).
- Error messages are displayed in the same region of the screen where graphs are drawn. An error message will go away if you click on it. Or, you can just fix the problem and the error message will go away the next time a graph is drawn.
- Never trust a computer! Do not assume that what you see in this program is perfectly accurate. For one thing, there are probably some bugs -- out and out programming errors. More important, the methods used in computations are approximations and in some cases guesses. Graphs are drawing by plotting a lot of points and connecting them with lines. Since a lot of points are used, generally the results are pretty good, but if the graph varies rapidly the results can be positively deceptive.
The Main Screen
The main screen is not the most interesting part of xFunctions! The more interesting "utility screens" are described below. However, you have to use the main screen if you want to define new functions to use in the utility screens.
The main screen shows a list of available functions on the left. Click on a function to see its graph. To add to this list, click on one of the buttons "New Expr.", "New Table", or "New Graph" in the lower left corner. This will take you to a screen where you can input a function. These input screens are describe below. If the function that is hilited in the list is a user-defined function, as opposed to one of the built-in functions, then the "Edit" button is enabled. If you click the "Edit" button, it will take you to the appropriate function input screen, where you can modify the definition of the function.
If you click on the graph, a cross-hair will appear marking a point on the graph. The coordinates of the point will appear under the graph. There is also an input box under the graph where you can input an x-value. This gives you much finer control of the x-value than you can get by clicking on the graph. Either press return in this box or click the "Set" button, and the corresponding point will be marked on the graph. If you click-and-drag on the graph, the cross-hair will move as you drag the mouse.
Right-click on the graph to zoom in on a point. Click with the middle-mouse button, or hold down the ALT key and left-click, to zoom out. You can also control the ranges of values displayed on the x- and y-axes with the controls along the right edge of the screen.
The main point of xFunctions is its seven "utilities." Each utility is a screen that allows you to work with functions in a different way. To go to a utility, select its name from the pop-up menu at the top of the applet. This pop-up menu is visible on the Main Screen and on each of the utility screens, so it is easy to move among these screens. (The pop-up menu is not available on the function input screens.) On other Web pages that use xFunctions, there might also be "examples" in this pop-up menu. Selecting one of the examples will take you to one of the utility screens and load data for some example into that screen.
Below, you'll find brief instructions for each of the seven utilities. However, much of the interface for the utilities is pretty self-explanatory.
Expression Function Input Screen
When you click the "New Expr." button on the main screen, you will be taken to an input screen where you can define a function as an expression such as "x^2+2*x+1". You can also define split functions, which have different expressions on different parts of their domains.
To define a function, set the name of the function in the input box the top of the screen (or just accept the name that is already provided there). To define a function using just one expression, just fill in the box labeled "Let y =". If you want to limit the domain of the function, you can enter a condition such as "x > -1 and x < 1" in the box labeled "provided:". The condition can use the relational operators =, <>, <, >, <=, and >=, and it can use the logical operators "and", "or", and "not". You can use any expression as part of the condition, such as "abs(sin(x)) < 0.5". You can define a function by up to four cases by filling in more of the boxes labeled "or y=" and "provided". For example: Let "y = x^2" provided "abs(x) < 1" or "y = x^3" provided "x >= 1" or "y = 1" provided "x <= -1".
Click the "Test" button to see the graph of your function. The limits on the axes here do not limit the domain of the function. They are only used to determine what part of the graph will be shown by default on the Main Screen.
Click the "Done" button to define the function and go back to the Main Screen. You must click either the "Done" or "Cancel" button to get out of the Expression Function Input Screen.
Graph Function Input Screen
Functions don't have to be defined by expressions! They can also be defined by graphs and by tables of values. From the Main Screen, click the "New Graph" button to go to the Graph Function Input Screen, where you can define a function as a graph. Technically, the graphs used here are Bezier functions, which are made up of segments. Each segment is a piece of a cubic polynomial, which is completely determined by the two points at the ends of the segments and by the slopes at those two points. It is possible to make graphs that have discontinuities, corners, and cusps.
A new function is just a line segment on the x-axis, with a point at each end. You can double-click the graph to introduce new points. Each point has a "handle", shown as a black point nearby. You can click-and-drag the points and the handles to modify the graph. Other operations are described in the "Brief Instructions" at the top right of the screen. If you want fine control over the x-value of a new endpoint, enter the x-value in the box labeled "at x =" and either press return or click the "Insert New Point" button. Note that the x and y limits in the lower right do not affect the shape of the graph, just how the xy-values on the graph are interpreted.
Click the "Done" or "Cancel" button to return to the main screen.
Table Function Input Screen
Use the Graph Function Input Screen if you want to set the general shape of a graph by hand. If you have a list of xy-points on the graph, enter them as a table in the Table Function Input Screen. You get to this screen by pressing the "New Table" button on the Main Screen.
Just enter the pairs of xy-values one-by-one in the boxes labeled "Input x" and "Input y". (You can press tab after entering the x-value, and press return after entering the y-value.) The points will be shown in a list in the upper right corner of the screen. If you click on a point in the list, it will be copied to the input boxes, so you can edit it. There is a button for deleting the selected point from the list.
Just knowing some points on the graph doesn't tell you what the whole function looks like. You have to specify what happens between the points. You have three choices: A "Step Function" has a constant value between two consecutive points from your list. For a "Piecewise Linear Function", the points are connected by line segments. A "Smooth Function" is continuous and differentiable at all points. (It's actually a Bezier function, defined by cubic polynomials between the points specified in your list.)
A preview of the graph is shown in the lower right (but not until there are at least two points). The limits on the axes are increased if necessary, and or can set them yourself.
Click the "Done" or "Cancel" button to get back to the Main Screen.
The Multigraph Utility Screen let's you draw up to eight functions in different colors on the same set of axes. You can select the number (1 to 8) of the function to be graphed with a pop-up menu near the lower left of the screen. Each curve is a different color, shown in a colored patch next to the pop-up menu. Enter the function and press return (or click "Graph!") to graph the function. Right-clicking on the graph will zoom in on a point. Click with the middle mouse button, or hold down the ALT key and left-click, to zoom out.
The Animate Utility Screen lets you work with a function whose definition can contain the parameter k, as well as the variable x. This is really a family of different functions, one for each value of k. The Animate Utility plays a "movie" showing a sequence of the graphs for different values of k. There are input boxes where you can specify the values of k at the start and end of the movie. You can also specify the number of intervals in the movie. (The number of frames is the number of intervals plus one.) What happens when you get to the end of the movie? The Animate Utility can either jump directly back to the beginning, or it can play the movie backwards. Which it does is controlled by a checkbox labeled "Loop Back and Forth." The "Go" button will start the animation. The "Next" button will move the animation from one frame to the next. You can use the mouse to zoom in out on the graph.
Parametric Curves Utility
The Parametric Curves Utility draws parametric curves in the xy-plane. A parametric curve is a plot of xy-points where the x and y coordinates are functions of some parameter. In this utility, the parameter is t. For example, the parametric curve defined by x=cos(t), y=sin(t) for t between 0 and 6.3 is a circle. (6.3 is about 2*pi.)
In this utility, you can enter x and y functions for up to eight parametric curves. There is a pop-up menu near the bottom left corner of the screen that you can use to select the curve whose definition you want to enter/display. Each curve is a different color, shown in a colored patch next to the pop-up menu. There are input boxes for specifying the range of t-values to use for the curves. You can also specify the number of points on each curve. (Remember that xFunctions draws graphs by plotting points and connecting them with lines. The results for discontinuous or rapidly changing functions might not be great. You might get a better curve by increasing the number of points.)
There is a button labeled "Trace". If you click this button, a crosshair will be moved along the curve whose number is specified in the pop-up menu. This lets you see how the point on the curve varies as t changes. The crosshair disappears after the curve has been traced once.
Zooming with the mouse will work in this utility.
The Derivatives Utility will draw the graphs of a function, the first derivative of the function, and the second derivative of the function. The three graphs are displayed in separate panels in the bottom half of the applet. If you left-click and drag on the graph of the function, a tangent line to the graph will be displayed. At the same time, the corresponding point on the graph of the first derivative will be marked. (You can also click and drag on the other graphs and see what happens.) Mouse zooming will also work in this utility.
Riemann Sums Utility
The Riemann Sum Utility computes Riemann sums for a specified function on a specified interval. The values of various Riemann sums, using different rules, are shown at the right. (The trapezoid rule isn't strictly a Riemann sum, but it's there as a choice anyway.) The area corresponding to one of the sums is displayed on the graph. Use the pop-up menu at the lower right to select which area is displayed. If you click on the graph, you get details about one of the rectangles or trapezoids in the sum. (Click on the blue information box to get rid of it.) Note that for the "~Circumscribed" and "~Inscribed" rules, the maximum and minimum of the function on a sub-interval is only computed approximately. (This explains the "~", which stands for "approximately.") Mouse zooming does not work on this screen.
Integral Curves Utility
The Integral Curves Utility draws direction fields and integral curves. An integral curve is the path followed by a point whose velocity at each point is given by a "vector field". A vector field on the xy-plane is given by a pair of functions of x and y. For the curve, these functions specify the derivatives dx/dt and dy/dt at each point. In the Integral Curves Utility, you can specify the functions dx/dt and dy/dt, and you can "drop" points onto the xy-plane and see the curves along which they move. The utility also draws a grid of arrows. An arrow at a point shows the direction of the integral curve as it passes through that point. (This is a "direction field" rather than a "vector field" because it shows only the direction of the curve, not the speed.)
You can start a curve by clicking at a point on the graph. Alternatively, for finer control, you can enter the x and y values of the starting point of the curve in the boxes labeled "Start x" and "Start y". Then, click the "New Curve" button to start a curve at that point. You can have as many curves as you want. If you turn on the "Extend curves in both directions" option, you get to see how the point would move backwards in time as well as forwards.
An integral curve is drawn by looking at the current position of the point and at the derivative functions at that point (and possibly at some nearby points for greater accuracy). This is used to project the point forward "dt" time units into the future. This is only an approximation! A smaller value of dt will generally give a better approximation. There is an input box where you can specify the value of dt. The Integral Curves Utility can use three different approximations: Euler's Method, Runge-Kutta Order 2, and Runge-Kutta Order 4. The utility has a set of radio buttons that you can use to select the method. Runge-Kutta Order 4 is the most accurate method, and there is really no reason to use the other two unless you specifically interested in the method rather than in getting the best possible approximation of the true integral curve.
Once a curve has been started, it uses the same method and the same value of dt forever. This lets you start different curves from the same point, using different methods or different values of t, to compare them.
In this utility, more than in others, you might want to use the "Equalize" button to equalize the scales on the x and y axes. Otherwise, the arrow directions, as drawn, can be misleading. You can use the mouse for zooming in this utility.
Graph 3D Utility
The Graph 3D Utility draws three-dimensional graphs of functions of two variables. The function to be graphed is of the form z=f(x,y), where f(x,y) is given as an expression that can contain variables x and y. The values of xmin, xmax, ymin, and ymax have a different interpretation in this utility: They specify a rectangle in the xy-plane that is used as the domain over which the graph is drawn. You can also enter zmin and zmax, which specify the range of z-values that you want to see. The scale on the graph is adjusted so that the specified ranges of x, y, and z are visible. But since the image on the screen is a two-dimensional projection of a three dimensional object, there is not an exact correspondence between the ranges and the boundaries of the rectangle shown on the screen.
To produce the graph, a rectangular grid of points in the domain is used. The function is evaluated at each point, and the resulting xyz-points are connected with lines. You can enter the grid size -- that is, the number of points along each side of this grid of points. The utility can draw graphs in four different styles: When only the lines between the points are drawn, the result is a so-called "wire-mesh model." The second style also consists of just lines, but lines that are hidden by other parts of the graph are not shown. Instead of drawing the lines, the regions between the lines can be filled in with color, giving a "shaded model." (The color used is determined by the orientation of the graph surface, so it looks as if the graph is illuminated from a certain direction.) In the fourth style, both the lines and the shading are shown. You can select among the four styles using a set of radio buttons under the graph. The shaded model without lines only really looks good on a high color monitor using a large grid size.
When the utility starts up, you will see a projection of the x-, y-, and z-axes along with a rectangle that frames the domain of the function in the xy-plane. When you press the "Graph it!" button or press return in any input box, the graph will be drawn. You get to see the graph being drawn, from back to front. (This is deliberate -- for understanding the graph, seeing the drawing process is more effective than just seeing the final result.)
You can rotate the axes and graph using two scroll bars. The horizontal scroll bar rotates the graph around the z-axis. The vertical scroll bar tilts the image towards you and away from you.
Mouse zooming does not work in the Graph 3D Utility. Do not expect Graph 3D to work well with most discontinuous functions.
David Eck (firstname.lastname@example.org), 27 October 1999