The mathXpert Calculus Assistant is the third part of a 3-tier advanced Mathematics series. Each successive title in the series includes all the content of the previous title and is therefore higher priced.

Part 3: Calculus Assistant, the product described on this page — includes all the material of Algebra Assistant and Pre-Calculus Assistant

MathXpert Series Overview

MathXpert is a system that allows a person to do mathematics on a computer screen in much the same way as it is done with pencil and paper, but with some important differences:

1. It is not possible to make a mistake. 2. If you do not know what to do, MathXpert can help you.

MathXpert is designed to provide a self-correcting way to learn mathematics. For example, suppose you think (incorrectly) that ln(x+y) = ln x + ln y. You will not be able to apply this incorrect rule in MathXpert, because only correct rules are offered as menu choices for you to choose. If you have in mind to use this incorrect rule, you will learn something from its unavailability. You may, of course, have to ask your teacher why it isn't there. MathXpert is meant to replace blackboards and homework (and graphing calculators), but not teachers and books.

The first and most important design principle of MathXpert is the "glass box" or "transparency" principle — show all the steps. Indeed, this principle combines with the idea that the user should be in control to dictate the basic design of MathXpert: the user chooses the steps and thus controls the development of the computation, while the computer carries out the low-level details.

Carrying out a calculation

To step through a problem, you use the mouse to select a term to work on, then you choose an operation from a short menu of possibly relevant operations. You do not have to type anything or memorize any commands. Since MathXpert carries out the operation for you, no mistake in the mechanics is possible. You can concentrate on choosing what to do. For example, to determine the value of x in the simple linear equation 3x + 2 = 11, when you use the mouse to highlight the 2, a menu pops up giving you a choice of operations that can be carried out, one of which is "subtract from both sides". When you select that, a new line is produced reading either 3x + 2 - 2 = 11 - 2 or 3x = 9, depending on your level (see below). If you didn't know what to do for that first step, clicking the Hint button would have popped up the suggestion, "Subtract something from both sides." If you still didn't know how to proceed, the Show Step button would have made MathXpert perform the correct task for you.

Working with a graph

You're never more than one click away from a graphical representation of any equation. The tools in the Graph Toolbar permit you to zoom out and zoom in (on the horizontal and vertical axes separately), to select a rectangle for the next drawing, to move a cursor (the Point and Slope tool) along the graph to display the numerical values of points on the graph and the slope at that point, and to change the values of parameters in the formula so that you can see how the graph changes if those values change. (Parameters are extra letters in the formula, for example b in y = x + b.) MathXpert, unlike any other software, uses its algebraic module and its prover to internally calculate the singularities of a function before graphing it. This makes it able to draw graphs that graphing calculators are incapable of drawing correctly.

Getting a problem to work with

You can select one of MathXpert's more than 1600 pre-stored problems, without having to type anything. Or, perhaps your teacher has prepared a homework file in MathXpert for you, so you can get your own homework without typing. Finally, if you want to enter your own problem, you can type it in. To start with, choose Edit on a similar problem provided by MathXpert, and you will see what form MathXpert expects you to type. Entering problems into MathXpert is easier than entering them into a scientific calculator. MathXpert never intervenes to provide help. You are free to "wander" by taking any mathematically correct steps at all, whether or not they are leading toward the "ideal" solution of the problem. Having reached an awkward solution, you may choose to compare it with MathXpert's automatic solution -- but that would be your choice. At no point will the program specifically tell you to do something different.

MathXpert's Algebra is correct

Many people think that computers would never make a mistake in algebra, but in fact this "correctness principle" is usually violated by computer algebra systems. For example, in most other computer algebra systems, you can start with the equation b = 0, and divide both sides by b. On the left you will get b/b = 1, and on the right you will get 0/b = 0. Thus you have derived the wrong equation 1=0. The problem here is that the other systems do not keep track of the assumptions and the side conditions. The side conditions of operations can be quite complicated. For example, if we try to integrate 1/x from -1 to 1, many computer algebra systems will simply form the indefinite integral 1/|x| and evaluate it from -1 to 1, producing the answer zero, failing to note that the integral is undefined due to the singularity at zero. The side condition that the function should be continuous on the (closed) interval of integration is not checked.

While scientists and engineers should be able to judge the correctness of answers they receive from computer algebra systems, that cannot be said for students. MathXpert satisfies the correctness principle by keeping track of assumptions and side conditions, and being able to infer side conditions from assumptions.

Instruction is tailored to the student's level

MathXpert contains a "user model" that consists of recording, for each of some 1600 mathematical operations, whether that operation is at the moment well-known, known, learning, or unknown. This controls the level of detail shown in the automode solution. For example, in the linear-equations example above, when subtracting something from both sides is learning, it does not simplify the result, but when it is known or well-known, it does simplify the result. Thus we got 11-2 in the first case and 9 in the second case. If this problem were entered under an advanced-algebra topic, it might have been solved in one step with the justification Solve linear equation. That one-step operation is not used in beginning algebra, because there it is classified as unknown. This illustrates how MathXpert tailors its output: it sets up an ideal user model based on the initial choice of topic. If the topic is elementary, the user model will set more operations to learning or unknown. If it is advanced, some powerful operations that are not used in elementary topics become available.

mathXpert Calculus Assistant features

As well as the ability to work through an unlimited number of your own problems, the mathXpert Calculus Assistant includes 7,000 pre-designed examples. They cover all the topics found in Algebra Assistant and Pre-Calculus Assistant, plus the following:

Limits (Calculus 1)

Limits of polynomials

Simple Limits

Limits of trig functions

Limits of rational functions

Limits at infinity

Infinite limits

Further Limits (Calculus 2)

Limits of exponentials

L'Hôpital's rule

Limits using leading term

Limit review

Differentiation (Calculus 1)

Differentiate from definition

Differentiate polynomials

Basic differentiation rules

Differentiate trig functions

Chain rule

Differentiation review

High-order derivatives

Further Differentiation (Calculus 2)

Differentiate from definition

Differentiate exponentials

Differentiate logarithms

Logarithmic differentiation

Differentiate inverse trig functions

Differentiate hyperbolic functions

Differentiation review

Integration (Calculus 1)

Sigma notation

Integrate polynomials

Simple integration

Fundamental theorem

Integration by substitution

Integrating by parts

Integration review

Further Integration (Calculus 2)

Integration by parts

Integrate to logarithms

Trigonometric integrals

Trig substitutions

Integrate rational functions

Rationalizing substitutions

Integration review

Improper integrals

System Requirements

Win 98/ME/2K/NT/XP — can be run on a Mac using Virtual PC or on Linux using WINE.

HTML Browser required. (Internet Explorer 5.0 or later recommended.)

The software is available in different license versions:

Single User

Site License: A single-room lab up to 35 computers, plus a single teacher's copy.

This software is a web download. We will send you a unique serial number you can use to download and install the software.

Please select the license version you require from the drop-down menu below. The price increase over the single user edition is displayed next to the license version you select.