Learn some of the ins and outs of Maple or Mathematica. This overview is designed for beginners and intermediate users, highlighting many of the salient features of the software. Understanding these features will help you become more effective and efficient in using Maple or Mathematica for analyzing mathematical formulations and solving a wide variety of problems. Topics include: general information, commands, lists, assignments, making your own commands, parts of lists, rules, retrieving output, symbol definitions, and pattern matching. As you become more proficient in using Maple or Mathematica, we recommend that you periodically revisit this overview to deepen your understanding of how the software works.
(Thomas’ Calculus, 10/e: Preliminary Chapter — Section 7, Exercises 2, 4, 6, 8, 9 and 10)
Mathematical models help us better understand things in the world around us, like the behavior of springs, safe driving practices, radioactivity in medicine, the growth of trees and fish, and the biology of mammals. In this module you will learn how to construct mathematical models, how analyze and improve them, and how to use them to learn about the thing you are modeling and to make predictions. As you will see, computer algebra systems like Maple and Mathematica are powerful tools that will aid you in your mathematical modeling.
(Thomas’ Calculus, 10/e: Chapter 1)
In this module, you will explore some limits by graphing the functions involved and also by creating tables of values for the functions.
(Thomas’ Calculus, 10/e: Chapter 1 - Section 1.5, Chapter 2 — Sections 2.1, 2.2, and 2.4)
To design high-rise buildings for seismic loads, structural engineers must be able to completely describe the motion that occurs during an earthquake. As a high-rise structure moves back and forth, the motion of any single floor is along a straight line, and can be completely characterized by knowing the position, velocity, and acceleration as functions of time. This module includes three specially designed Maple or Mathematica commands that present dramatic animated visualizations of the derivative relations among the position, velocity, and acceleration. In addition to the seismic vibration of each floor in a building, a variety of other motions are presented including constant velocity, constant acceleration, harmonic oscillation, and decaying oscillations. You can also use the specialized Maple or Mathematica commands to study other motions that are of interest to you.
(Thomas’ Calculus, 10/e: Chapter 2, Section 1)
This module contains a specially designed function that can be used for demonstration and exploration. The function derivApprox extracts n evenly space sample points from a differentiable function and calculates the slopes of the secant lines between consecutive pairs of sample points. It then plots the function, the sample points, and the secant lines on one graph. A second graph is plotted to show the slopes of the secant lines in comparison with the derivative of the function. As the number of sample points increases, the secant slopes get closer to the derivative function. You are invited to adapt the Maple/Mathematica code for derivApprox to generate a list a values to approximate the second derivative and plot the approximations superimposed with the second derivative function.
(Thomas’ Calculus, 10/e: Chapter 3, Sections 1 through 3, and Chapter 4, Sections 1 and 2)
The design requirements and specifications require the structural engineer to ensure that beams are strong enough so that they won’t fail under the loads that they have to support. In addition, to prevent damage to windows, ceilings, and wall partitions, maximum deflections of beams under applied loads must be limited. In this module you will learn how engineers use calculus to accomplish these tasks when designing a structure.
(Thomas’ Calculus, 10/e: Chapter 3, Sections 3.1 through 3.3 and 3.4, and Chapter 4, Sections 4.4 and 4.5)
What do flood control, riding an elevator, and launching a model rocket have in common? We can describe all of these, using calculus. In addition to gaining a better understanding and appreciation for the Fundamental Theorem of Calculus and the Mean Value Theorem, you will also see the importance of finding the areas between the graphs of functions, and identifying extreme values. The purpose of this module is to show the wide applicability of some of the basic ideas in the calculus of single variable functions. Let's get started.
(Thomas’ Calculus, 10/e: Chapter 3 — Section 7, Problem 20.)
To how many place can you estimate p or e
or 
? In this module, Isaac Newton and
Maple/Mathematica team up to help you out in this endeavor. Estimates to thousands
of places are no sweat at all when you have such an awesome pair to help you.
(Thomas’ Calculus, 10/e: Chapter 4, Sections 3 and 4, and Chapter 5, Sections 3 through 5)
Areas under a curve, volumes of revolution, arc lengths of smooth curves, and surface areas of shells of revolution are estimated by formulating difference equations that approximate the accumulation of the various quantities of interest. The numerical solutions of the difference equations give approximations of each specific quantity. The process is then reversed to produce a sequence of differences from the solutions of the difference equations, bringing the analysis full circle.
Velocity Position
(Thomas’ Calculus, 10/e: Chapter 4 - Section 4.6 and 4.7)
This module includes two specially designed Maple/Mathematica commands that present dramatic animated visualizations of the integral relations between the acceleration, velocity, and position of an object moving along a straight line. A variety of motions are studied including constant velocity, constant acceleration, harmonic oscillation, and decaying oscillations. You can also use the specialized Maple/Mathematica commands to study other motions that are of interest to you.
(Thomas’ Calculus, 10/e: Chapter 6, Section 4, #21 and Chapter 8, Additional Exercises, #37)
In this module you will see how a mathematical model can be used to determine a safe and effective drug-dosage regimen that maximizes patient comfort. Certain drugs are ineffective when the concentration in the blood drops below a lower effective threshold value, and they can be harmful to the patient if the concentration exceeds an upper safe threshold value. An initial loading dosage is followed by maintenance dosages at prescribed time intervals or by a continuous IV (intravenous administration). The model is a simple first-order differential equation with the discrete drug-dosage regimen constructed using a special mathematical function called the Dirac delta. See if you can determine a drug-dosage regimen that keeps the concentration in the blood between the safe and effective levels while minimizing patient discomfort.
(Thomas’ Calculus, 10/e: Chapter P, Section 7, and Chapter 5, Section 4, Exercise #24)
In a recent television commercial, a bungee jumper comes so close to the ground that he is able to dip a corn chip that he holds in his mouth into a bowl of salsa on the ground without bumping his nose. (Note: in the commercial it actually was a "he" but it could just as well have been a "she".) In order to do this successfully, the bungee jumper (he or she) would have to do some serious mathematical modeling. So, in order to present my students with "The Bungee Cord Challenge", I went to a local hardware store and purchased some bungee cord material (approximately one-eighth inch in diameter), cut it into various lengths, and the gave a piece to each group of students in my class. I also gave them a set of 0.2 kg masses and a metric tape measure. Here is the challenge I presented them with: Given a fall distance, how long should you cut the bungee cord so that a 0.2 kg mass, when dropped from that height, would just "kiss" the floor? In this module, you build a model (with your own data if you wish), test the model, use it in calculations, make predictions, and test the results.
(Thomas’ Calculus, 10/e: Chapter 7, Sections 5 and 7)
How can you use a game of chance to evaluate an integral and what do games of chance have to do with improper integrals? This module will give you insight into both of these issues.
- You will generate random points within a fixed region and then estimate the area of the desired portion by considering the percentage of random points that fall within the boundaries of the desired portion.
- You will consider a probability density function defined over an infinite interval and explore the computation of means, moments, and probabilities associated with the density function.
(Thomas’ Calculus, 10/e: Chapter 8, Sections 9 and 10)
This module contains a specially designed function that can be used for demonstrations and exploration. The function taylorpolydemo demonstrates the convergence of Taylor polynomials to a function that has derivatives of all orders over some interval of its domain. It generates a series of plots showing the function and the Taylor polynomial as the degree of the polynomial increases. This series of plots can be animated to give a dramatic demonstration of the convergence of the Taylor polynomials. Several functions are included in the module and others can easily be added. If you are interested in how the taylorpolydemo function works, a second part of the module leads you through the construction of the function with detailed explanations.
Note about the next two modules that deal with the Fourier series:
It should be noted that Mathematica has a package that allows you to merely call upon the Fourier Series. The package is called upon by executing <<Calculus`FourierTransform` The specific command for getting the Fourier Series expansion is different in Version 3, as compared to Version 4. You can get details on this from the Help window, by looking under Add-Ons and then Calculus packages and then the FourierTransform. At present, this module does not call upon this package, but simply uses the Integrate commands to compute the Fourier coefficients.
(Thomas’ Calculus, 10/e, Chapter 8, Section 9 — reference to #’s 5 and 15)
Did you know that the Fourier series can be used to recreate musical tones? Do you want to let the computer do the tedious part of computation with the Fourier series and be able to visualize the results? This module leads you through different perspectives on the Fourier series, including even and odd functions, piecewise continuous functions, and also discrete functions.
(Thomas’ Calculus, 10/e: Chapter 8, Section 10 — reference to #’s 8 and 16)
This module leads you through use of the Fourier series with even and odd functions that are also piecewise continuous functions.
(Thomas’ Calculus, 10/e: Chapter 9 - Section 3 and 5)
Parametric equations are very powerful and the purpose of this module is to help you get used to the idea of expressing curves using parametric equations and analyzing motion in the plane using these parametric equations. Polar plots can also be expressed parametrically and that form lends itself well to easy translation into Cartesian coordinates.
(Thomas’ Calculus, 10/e: Chapter 10, Section 3, #61 and Extension)
A problem that artists have faced since time immemorial and that computer graphics persons are faced with today is that of picturing three-dimensional objects on a two-dimensional plane. Depending on the position of the "eye of the beholder," the representation may vary. In the project that follows, two aspects of this problem are analyzed. Both are linear mappings of points in three-space to a plane. Part 1 is the problem suggested in the text (#73 from Section 10.3) where the viewpoint is from a single beholder. The remainder of the lab focuses on parallel projections, similar to the situation when X-Rays are taken and the need for CAT-Scan technology is demonstrated. The linear algebra behind such mappings is introduced in Part 5.
(Thomas’ Calculus, 10/e: Chapter 10, Sections 3 and 4)
In this module, you will learn how Maple/Mathematica can help you plot the lines, planes, cylinders, and quadric surfaces you have been reading about. In the process of plotting lines and planes in 3-dimensions, you will gain insight into their vector definitions.
(Thomas’ Calculus, 10/e: Chapter 10, Sections 5, 6 and 7)
Motion in three-dimensional space is something with which we are all familiar, but visualization of equations of motion and computations involved in analyzing that motion can become very tedious. This module gives a broad overview on the analysis of motion where the equations are given parametrically.
(Thomas’ Calculus, 10/e: Chapter 11 - Section 11.1 and beyond)
In this module, you will learn how Maple/Mathematica can help you plot functions of two variables. For the standard form, z = f(x,y), you will examine contours (z-level curves) and x-level and y-level curves. Can you come up with a function that will stump your classmates when they attempt to match your functions with your surface plots, your contours, and your level curves? This is a challenge to you to explore non-trivial functions and their graphs. All the codes for creating the necessary graphs are in this Maple/Mathematica notebook.
When functions of two variables are expressed in cylindrical or spherical coordinates or are expressed parametrically, you can read in a Maple/Mathematica package that will enable you to plot these surfaces.
(Thomas’ Calculus, 10/e: Chapter 11, Section 5)
What does the directional derivative mean and what does it look like? You get to put yourself in the position of a skateboarder and explore skating on a ramp or on a parabolic bowl. The graphical concept of dotting the gradient of the surface function into a unit vector in the direction of motion in the domain will become clearer and you will be able to plot the directional derivative as a function of the parameter t.
(Thomas’ Calculus, 10/e: Chapter 11, Section 7, #45)
We look for patterns in real data sets including medical, socio-economic, meterological, and pollution statistics. Does the least squares method apply only to finding lines of best fit or could you apply the technique from scratch to any curve you wish to fit to a set of data points? We begin by analyzing data to which standard fit functions might apply and then utilize the calculus definition of best fit to find a non-standard fit function.
(Thomas’ Calculus, 10/e: Chapter 11, Section 8)
How do you maximize or minimize a function that is subject to constraints? The skateboarder from the directional derivative project returns to determine precisely where along the figure eight traversed the high and low points of the surface are reached. You will investigate what this has to do with the directional derivative.
(Thomas’ Calculus, 10/e: Chapter 11, last part of end of chapter exercises)
Do you want to see physical interpretations of the contours and level curves you have been drawing. This is what the exploration of the heat equation will do for you. The heat equation is the solution to a partial differential equation and is a meaningful way to employ the concept of the Fourier series.
(Thomas’ Calculus, 10/e: Chapter 12, Section 1)
How can you use a game of chance to evaluate a multiple integral? That is just what you will do with this project. You will generate random points within a fixed region and then estimate the volume of the desired portion by considering the percentage of random points that fall within the boundaries of the desired portion. Since this will only be an estimate of the true volume, you will explore the accuracy of this method.
(Thomas’ Calculus, 10/e: Chapter 12, Section 2)
What do means and multiple integrals have to do with probabilities? How can the method of moments be used to help determine whether or not an object will float in an upright position? These questions will be answered if you explore this module and you will also get to practice new and interesting ways to plot functions.
(Thomas’ Calculus, 10/e: Chapter 12, End of Chapter Exercises - #’s 33 and 34)
How can you measure precipitation if you collect rain in a container that is narrower at the bottom than at the top? To what angle must a satellite dish be tilted in order that rain not collect in it?
What Difference Do Conservative and Non-Conservative Force Fields Make?
(Thomas’ Calculus, 10/e: Chapter 13, Section 3 — includes # 30)
You will explore integration over vector fields and experiment with both conservative and non-conservative force functions and with different curves. This should assist you in understanding how to do line integrals, as well as, getting a better appreciation for situations when the work done is independent of the path taken.
(Thomas’ Calculus, 10/e: Chapter 13, Section 4 — includes # 34)
If you can visualize a force field and a curve on it, how can that help you in understanding the work integral? Does Green's theorem really work? In this module, you will explore integration over vector fields and use parametrizations to compute line integrals. You will also explore how you can determine the closed curve around which your work integral is a maximum?
(Thomas’ Calculus, 10/e: Chapter 13, Section 4)
If you can visualize a force field and a curve on it, how can that help you in understanding the flux-divergence integral? Does Green's theorem really work? In this module, you will explore integration over vector fields and use parametrizations to compute line integrals. You will also see if it makes any difference whether or not a force is conservative.
(Thomas’ Calculus, 10/e: Chapter 13, Section 8 - includes #24)
You will see that surface integrals are still difficult to set up and to evaluate, even with Maple/Mathematica, but parametrization can assist in evaluating flux surface integrals. You will also verify that the divergence theorem works, just as you have done with Green's theorem and maximize the flux through a closed region by carefully selecting the size of the region?
This module is intended as an example. There are no You Try It exercises, due to the complexity of the nature of the problems. However, the student who is quite familiar with Maple/Mathematica could adjust this code to solve another problem.