In 1988, Keith Geddes and others involved with the Maple project at the University of Waterloo published a Maple Calculus Workbook of interesting calculus problems and their solutions in Maple. Over the years, I've paged through this book, extracting some of its more unique problems. Recently, I extracted the following problem from this book, and added it to my Clickable Calculus collection, which I use for workshops and web-based presentations.

Three recent articles in the Tips & Techniques series addressed the question of stepwise solutions in Maple. Just what is it that Maple provides by way of stepwise solutions for standard calculations in the mathematical curricula? There are commands, assistants, tutors, and task templates that provide stepwise calculations in precalculus, calculus, linear algebra, and vector calculus. In addition, since Maple can implement nearly any mathematical operation, any stepwise calculation can be reproduced in Maple by assembling the appropriate intermediate steps, just as they would be assembled when working with pencil and paper.

I have to thank my friend John Wass, an editor from Scientific Computing magazine who began a recent article with the clear warning “Attention Engineers! The developers at Maplesoft rarely sit still for very long.” This was a comment on the thrilling speed that enhancements are flowing from the MapleSim pipeline. Although his quote refers to a MapleSim 3 article he wrote, I chuckled as the sentiment still rings true as my colleagues and I catch our breaths after the recent release of MapleSim 4. Yes, the engineering community has definitely taken notice that MapleSim, in such a short amount of time, is already making a big difference in the way we do and think about modeling.

Our solar system was created in three hours… well, at least that was how long it took for me to create a model of it in MapleSim. This process started out as an inquiry from a MapleSim user asking if MapleSim can be used to model planetary motions, through the use of Newton’s law of gravity. I view this kind of inquiry as both a challenge (any time someone asks “can MapleSim do such and such” it is an automatic invitation for us Applications Engineers to try it out J ), and an opportunity to learn new things.

While I am somewhat fascinated by astronomy (who isn’t dazzled by all of those pretty photos of various celestial bodies in the universe?!), I have never developed a keen interest in it. That can be partially attributed to the fact that I grew up in a city that never sleeps, which means serious light pollution (I didn’t realize how beautiful the night sky was and how bright the stars can be until my teenage years on a family camping trip… but that’s another story). The aspect of astronomy that I understand tells me that the law of gravity applies, to a certain extent, and that the magnitude of the numeric values that we are dealing with (for planetary motion simulation) is astronomical! So for me, these are the two key issues that will need to be addressed when creating a MapleSim model.

Sometimes the obvious escapes me, and it’s only due to some chance observation that I realize the same fundamental principles are everywhere.

A short time ago, I created a simple hydraulic network in MapleSim, and after experimenting with some of the parameters, found it gave the same behaviour as an electric circuit I’d modeled earlier.

Some calculus texts compute volumes of solids by the method of "slices" before they discuss the methods of disks and shells. On the other hand, there are texts that start with disks and shells, then throw in a few examples of slices. In any event, these calculations are supposed to be illustrations of how definite integration is an additive process. Unfortunately, students often get lost in the details of the individual examples, and fail to see that all these calculations are just demonstrations that definite integration is a process of addition.

Do an internet search on "Challenger Puzzle" and you will find descriptions and solvers for a puzzle that involves sums of integers from one to nine. Indeed, on a 4 × 4 grid where sixteen integers would fit, four are given, along with the row, column, and diagonal sums of the numbers not shown. The object of the puzzle is to discover the missing twelve numbers.

Unlike Sudoku, the digits can repeat. And unlike Sudoku, the puzzle can have multiple solutions. In fact, "There may be more than one solution" is explicitly stated below the directions, copyrighted by King Features Syndicate, Inc., that appear in my local newspaper, the Waterloo Region Record.

After years of dreaming, planning, scheming, and hoping, my family, as a single entity, finally made it to Asia for a holiday. In our region, school children typically enjoy a mid March “Break”. This usually means families packing up their minivans and driving to Florida or other warm places to help them forget the bleakness of the Canadian winter. This year, however, the spring winds took my family to Asia – Japan and Korea to be specific. I was born in Korea but moved to Canada in 1971 when I was 7 years old. And those of you who have glanced at my past posts know that I’ve been a frequent business visitor to Japan many times over the years. But a family trip to these dynamic places is a completely different experience. There is something remarkable about the discovery experience you get as a pure tourist where issues of punctuality and protocol disappear and you’re left with the experience in its most raw and engaging forms.

Recently, I received an email from a physics instructor asking for help in building a tool that would display the solution of the initial value problem

with the four parameters under the control of sliders. (Of course, we recognize that this equation governs the damped, driven linear oscillator, and that the request to endow its solution with sliders is in service of visualization of the change in the nature of the solution as the parameters vary.)

A few days ago I asked my wife what she thought was the most important invention of the last 100 years. Without pause, she responded “the credit card!”

I suppose that “important” is relative, but my answer is the transistor. As Wikipedia puts it, “The transistor is the fundamental building block of modern electronic devices, and its presence is ubiquitous in modern electronic systems.” Semiconductors are vital to everyday life because they have radically reduced the size, cost and power consumption of all modern electronics.

Life wouldn’t be the same without transistors (or without credit cards, admittedly). Your laptop would weigh hundreds of pounds and be a “mainframe” computer without transistors. Your television would rely on tubes, and you couldn’t mount it on the wall – it would weigh 100 pounds or so without transistors. No mobile phones, either – your telephone would have a rotary dial, and the handset would be connected with a wire. Telephone directories would still be printed books weighing 5 pounds or so.

It was years since I "derived" the result that slopes of perpendicular lines were negative reciprocals of each other. So I thought it would be easy to show that when , where, in Figure 1, is the slope of line (black) and is the slope of line (red). Clearly, lines and are perpendicular when .

I’ve written in the past of how the push for more efficient, “greener” designs are driving innovation in important industries like auto, aerospace, and power. Over the past few years, we’ve met countless engineers around the world who are working hard to transform conventional designs to highly refined optimal designs in tune with modern realities, and some are, of course, throwing out old ideas all together and venturing into exotic power sources and radical platforms that used to be the stuff of science fiction. Last week I had one of the more interesting and enjoyable encounters with such a group of very talented green engineers.

Recently, I had to write a brief introduction to the precalculus topic "Vertical Translation of Graphs." Figure 1 ( in black, in red) says just about everything.

Figure 1 The red curve () is the black curve () vertically translated upward by one unit.

But is the issue all that trivial? Although the curves are vertically separated by one unit, they don't look uniformly spaced. The animation in Figure 2 helps overcome the optical illusion that makes it seem like the black curve bends towards the red curve, even though the curves are congruent.

Ten years ago, I wrote an article for Dr. Dobb’s Journal on Analytical Computing. Many of the techniques I discussed there, like hybrid symbolic-numeric computing and automated code generation have since revealed themselves as indispensable tools for engineering. Others, like exact computing, have yet to reveal their potential.

A lot has happened since that article, of course, and it’s about time I share some thoughts about what the current challenges are. There are three areas that are top of my mind and that I would like to discuss here: Parallel computing, collaborative software and user interface abstractions.

I’ve always been a big fan of languages and even a bigger fan of those who readily master multiple languages with relative ease. My late brother was a linguist with a minimum of five or so distinct languages in his portfolio. Yes, there were many things that I thought I could do better, but that one gift of his was the thing that I would remember him by as time went on. The other day, my son Eric asked me for advice on what courses to take in Grade 10. He essentially had three electives and, as with most public schools in our country, there were countless choices, all of which sounded tantalizingly interesting and enriching. In the end he came to the conclusion (OK, I drove him to the conclusion), that French, German, and Computer Science would be the right choices.

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