P.8 Technology Used in Calculus

Whether your instructor requires you to use a graphing calculator or a computer algebra system (CAS) or believes that calculus is learned best in the classical way, by hand, real applications of calculus—in engineering, science, economics, and statistics—will usually require the use of technology.

This text, as you see it, would not have been possible without technology. All the figures in the text were produced using technology—some on a graphing calculator, most with an interactive graphic system, others in Adobe Illustrator\(^{\circledR}\). The equations and symbols were created and spaced by a computer using the MuPad CAS; the page numbering and printing were done electronically.

In this brief section, we outline some of the more popular technologies currently used in learning calculus.

Since the 1960s, portable computation devices have been available. Their introduction eliminated the need to perform long, tedious arithmetic calculations by hand. Many calculations that were previously impossible can now be done quickly and accurately.

Many calculators today, certainly those you are using, are not so much calculators but small, hand-held computers. They numerically manipulate data and mathematical expressions.

Most graphing calculators have the ability to:

• graph and compare functions of one variable.
• graph functions of one variable in several formats: rectangular, parametric, and polar.
• locate the intercepts, local maxima, and local minima of a graph.
• graph sequences and explore convergence.
• solve equations numerically.
• numerically solve a system of equations.
• find a function of best fit.
• find the derivative of a function at a particular number.
• numerically approximate a definite integral

As you read the text, you will see examples of graphs generated by a graphing calculator. You will also find problems marked with , which alerts you that graphing technology is recommended.

In contrast to a calculator, a CAS symbolically manipulates mathematical expressions. This symbolic manipulation usually allows for exact mathematical solutions, as well as for numeric approximations. Most CAS systems are packaged with interactive graphing technology that can produce and manipulate two- and three-dimensional graphs.

There are many computer algebra systems available. They vary in versatility, ease of use, and price. Here, we outline the capabilities of several systems used in many colleges and universities.

Maple\(^{\rm TM}\) was developed in 1980 by the Symbolic Logic Group at the University of Waterloo in Ontario, Canada. It was the first CAS to use standard mathematical notation, and its source code is viewable. Maple\(^{\rm TM}\) is now owned and sold by Maplesoft\( ^{{\rm TM}}.\) The latest version, Maple\(^{{\rm TM }}\)17, was released in April 2012. Maple\(^{\rm TM}\) is marketed to mathematics educators, mathematicians, engineers, and scientists.

Maple\(^{{\rm TM }}\)17's “Clickable Math” allows the user to enter mathematical expressions into the equation editor in standard mathematical notation using keystrokes, menus, and symbol palettes. Operations can be initiated using context-sensitive menus, and the output is annotated for future reference.

Maple\(^{{\rm TM }}\)17 can be used to:

• generate two- and three-dimensional graphs using dropdown menus.
• resize, recenter, and rotate graphs using a mouse, and to overlay more than one graph on a set of axes.
• symbolically find limits, derivatives, and integrals with exact answers.
• numerically approximate limits, derivatives, and integrals.
• solve differential equations.
• perform dimensional analysis.

Maple\(^{{\rm TM }}\)17 also includes step-by-step calculus tutorials and a programming language that allows the user to write programs and perform analysis.

Mathematica was developed in 1988 by Stephen Wolfram of Wolfram Research, Champaign, Illinois. It is probably the most complete computer algebra system available. Written in C, Mathematica is a computational software program that allows mathematicians, engineers, and scientists to compute symbolically, visually, and numerically to any precision. The current version of Mathematica, Mathematica 9, was released in 2013.

Mathematica 9 features a free-form linguistic input that needs no knowledge of syntax. Mathematica 9 can be used to:

• solve linear and nonlinear optimization problems.
• find limits, derivatives, and integrals, both definite and indefinite.
• solve differential equations.
• generate two- and three-dimensional graphs.
• rotate and resize three-dimensional graphs.
• analyze huge data sets.
• explore formulas and solve equations.
• analyze mathematical functions.

Wolfram|Alpha (http://www.wolframalpha.com/) is a Web-based derivative of Mathematica, developed by Wolfram Research in 2009. It is called a computational knowledge engine, and its aim is to organize data. Since it is written using Mathematica, it can generate graphs, solve equations, and perform calculus.

MATLAB\(^{®}\) (short for matrix laboratory) is a technical computing language that supports vector and matrix operations and allows for object-oriented programming. MATLAB\(^{®}\) was developed at the University of New Mexico by Cleve Moler in the mid-1970s. Moler wanted to provide students access to matrix software without having to write a Fortran program. In 1984 Moler partnered with Jack Little to form the MathWorks\(^{\rm TM}\), Inc. After adding an interactive graphing system, sales of MATLAB\(^{®}\) grew. MATLAB\(^{®}\) is now used in such diverse products as cars, airplanes, cell phones, and financial derivatives. When packaged with the Symbolic Math Toolbox, MATLAB provides symbolic and numeric computing and extensive graphing capabilities. The current version of MATLAB\(^{®}\) was released in March 2013. MATLAB\(^{®}\) with the Symbolic Math Toolbox package can be used to:

• perform computations that require exact control over numeric accuracy to any number of digits.
• perform symbolic mathematical operations, including finding symbolic derivatives, integrals, and limits.
• interactively evaluate Riemann sums.
• find sums and products of series.
• perform Taylor series expansions.
• produce polar and parametric curves.
• produce contour and mesh surfaces.
• create animated two- and three-dimensional graphs.

The Symbolic Math Toolbox also provides users with access to the MuPad language for operating on symbolic mathematical expressions, extensive MuPad libraries in calculus and other areas, and the MuPad notebook interface with embedded text, graphics, and typeset mathematics.

MuPad (Multiprocessing Algebra Data Tool) is a CAS and high-precision decimal arithmetic program, as well as an interactive graphing system. It was developed in 1990 at the University of Paderborn, Germany. MuPad's syntax is modeled on the Pascal programming language and is similar to that used in Maple, but MuPad supports object-oriented programming. In 1997 MuPad was sold to SciFace Software GmbH \(\&\) Co. KG. Although it is no longer sold as a stand-alone, MuPad is the CAS that drives the Symbolic Math Toolbox of MATLAB\(^{®}\) and is the CAS used in Scientific WorkPlace\(^{®}\).

Sage (www.sagemath.org) is a free open-source CAS system that combines other open-source mathematics packages into a unified package written primarily in Python. Licensed under the General Public Licence, its stated mission is to “create a viable free open source alternative to Magma, Maple, Mathematica, and Matlab." The Sage project was begun in 2005 as a specialized system for number theory, and it continues to be developed. Sage can either be downloaded onto a computer or used through a Sage Network account. Although to use Sage requires entering code, there is a dropdown toggle that explains the code.

Until recently, only computers could support a CAS, but in the last decade or so, portable hand-held computer algebra systems have become available. These CAS look and work like calculators, but they compute symbolically and have enhanced graphic capabilities.

The TI-Nspire\(^{{\rm TM }}\) CAS, first released in 2007, is a hand-held system built with the Derive\(^{{\rm TM }}\) CAS. Derive\(^{{\rm TM }}\) was developed in 1988 by a software company now owned by Texas Instruments. Derive\(^{{\rm TM}},\) which is no longer sold independently, uses less memory than other CAS, so it works well in a hand-held device. The TI-Nspire\(^{{\rm TM }}\) CAS is used in schools and colleges because of its portability, affordability, and ease of use.

The Casio Prizm was released in 2010 and is another hand-held system used in many schools. Casio Corporation developed the first hand-held graphing calculator in 1985. In addition to the CAS, the Prizm includes applications to solve and graph differential equations.

The TI-Nspire\(^{{\rm TM }}\) CAS and the Casio Prizm can perform all the tasks of a graphing calculator. In addition, it can:

• obtain exact values in terms of variables \(x\) and \(y,\) and irrational numbers when performing algebra or calculus calculations.
• graph and rotate three-dimensional surfaces.
• find the derivative of a function.
• find an indefinite integral.

At appropriate places in the text, you will see problems marked , which alerts you that a computer algebra system is recommended. If your instructor does not require a CAS, these problems can be omitted. However, they provide insight into calculus and will enrich your calculus experience.