About Math

About Mathematics Commentary

Chapter 6. Calculus: The Smallest Pebble on the Beach

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What this chapter is about

This chapter is designed to give these students an overview of calculus with a focus on the remarkable connection between differential calculus (slope) and integral calculus (area). It is easy to dismiss this development as superficial, because limits are treated very informally, but it conveys the larger picture of the subject without the kind of details that these students do not need. For comparison with the continuous calculus concepts a brief introduction to finite math concludes the chapter. Note: This approach is far from new: it mirrors that of virtually all calculus texts in the first half of the twentieth century, the kind I studied when I was a college student. I find it interesting that, while the calculus was so truncated, analytic geometry was stretched into a full semester. How times have changed.

Although there are many more difficult areas of mathematics, calculus has a reputation of being a major academic hurdle. Because calculus has this reputation of being so difficult, I urge you to encourage your students as you work through the chapter with them. Consider offering comments like: “Now that isn’t so hard to understand, is it?” and “That’s all there is to it.” as you work through the chapter with them. A measure of your success with this chapter will be your students’ comfort with its ideas. Note: As I noted earlier, my own reward when I first taught this chapter came from my grader, a math major, who told me that, despite having studied calculus for three semesters, he had a better grasp of what calculus is about after following the presentation of this text.

The real trap that this chapter offers is the temptation to teach too much, to go into too much detail. The idea of the chapter is to give students only a general understanding. You should recognize, however, that this general understanding is an important contribution to their thinking.

Section Notes and Activities:

Section 6.1 is a quick and informal overview of limits without any reference to the usual epsilons and deltas. While it raises some of the problems with this informal approach to limits, it does not resolve them.

You could introduce and discuss two of Zeno’s paradoxes as further evidence of the problems with infinity:

You have already agreed in Section 4.3 to apply the formula S = a/(1—r) to resolve those situations, but does it really answer the paradoxes? Philosopher Kevin Brown says of them,

Section 6.2 is an introduction to slope. Because this is a topic at the heart of high school algebra, most students will be acquainted with it. One way to address it is simply to draw a slanted straight line on the board and ask students how to describe the appearance of that line to someone who does not see it.

A reasonable question: highways and railroads report grades in percents. What is the slope of a 1% grade?

Section 6.3 The transition from the slope of a straight line to the slope of a curved line displays the basic idea of the differential calculus, even though the graph is only a representation of this idea. I continue to believe that drawing a curve on the board and using a yardstick or pointer to represent a changing tangent to this curve is the best way to show students the concept that is being addressed. Then this same yardstick approach can be used to show how, for a given point, secants approach this tangent line as the distance between points on the curve decrease. As an aside, in the “old days” a piece of spaghetti and an overhead projector worked well with this too. Now there are a variety of websites which show visual images of this.

This approach becomes even more clarifying if you first do this for a general curve with no named function, but then do so for a specific curve for which you calculate the secant slopes as you show the yardstick secant.

And here is a perfect example of the role of a panel for Panel 6.3.1 does the calculations for the slope of those secants. Display a sketch of Figure 6.3.6 and hold your yardstick for different secants with the students using Panel 6.3.1 to give you its slope. All they need is to calculate delta x by subtracting x values of the two points.

Section 6.4 should represent a striking change for your students and one way you can stress this is by telling your students that what you address in this section was formerly taught in a separate course from the material in Section 6.3. In this way you lay the groundwork for the “remarkable connection” of Section 6.6. Note: If you look at catalogs from the mid-20th century you will find a semester devoted to Differential Calculus and another to Integral Calculus.

This section introduces the symbols ∑ and ∫ and carefully distinguishes between them. You probably recall one simple example of ∑ that has an interesting outcome. It is the sum of the first n odd numbers:

Having students use Panel 6.4.1 for various values of n will show (but not prove, of course) that the sum is n2.

The distinction between ∑ and ∫ that is made on page 144 is an important but subtle one and it is indeed glossed over there. It is, however, important to point out this difference. It will, of course, be partially clarified in the next section.

Section 6.5 does for the integral calculus what Section 6.3 did for the differential calculus and it is worth pointing this out to your students. If students understand the ideas in these two sections and see how Section 6.6 unites them, then they have a basic understanding of what calculus is all about. It may be bare bones, but this is it.

Just as Panel 6.3.1 carried the calculating weight in Section 6.3, Panel 6.5.1 does so here. I cannot stress how important this is. Surely all of us who have taught calculus to beginners have displayed figures like Figure 6.5.5 and calculated the area for a few rectangles — one, two, perhaps even three or four of them — and left it at that. Here you can do far better. After working out values for one and two rectangles and comparing answers with the panel, you can gather values for more numbers with the panel doing the calculations and approaching the 9 1/3 area.

Section 6.6 connects integration with differentiation, two concepts that, until Newton and Leibniz drew them together and applied them, had separate histories. The history of the integral calculus, for example, can be traced all the way back to Archimedes more than 22 centuries ago. The contentious battle for priority with Newton accusing Leibniz of plagiarism, that attended this late 17th century development is a topic worth discussing with your students. There are a number of associated questions: How do you decide what was discovered? How do you pin down a date? Why does priority matter? How do we today determine priority — do the methods differ between science and the humanities? Notes: The Fundamental Theorem of Calculus was actually developed a few years before Newton and Leibniz made their contributions by Barrow and Gregory. The contributions of Newton and Leibniz were far more extensive.Tom Lehrer’s song “Lobachevsky” could entertain your students here as it does in Chapter 10. You can access it from YouTube by searching Tom Lehrer Lobachevsky. An excellent book on this controversy is The Calculus Wars by Jason Sacrates Bardi.

The Newton-Leibniz controversy also caused problems because of notation differences, some of which confuse students still today. For example, we have all these different notations for the derivative: Dxy, f ’(x), dy/dx, and even y with a dot over it. (A well as the use of differentials, which happily we can avoid here.) These differences were exacerbated by adherents. You might ask students to compare this with today’s politics.

Section 6.7 is a very brief introduction to the indefinite integral, which is useful for undoing differential relationships — essentially by working backwards through them. For example, the distance to velocity to acceleration equation series is here reversed. Ask your students why the constant is not required in definite integration.

Section 6.8 suggests some situations when you might find calculus applied. You might ask your students to identify where these concepts might apply to their own lives or the subjects in which they are interested. If you do this you might consider their examples associated with finite rather than continuous mathematics and thus better connected to the next section. You can point out, however, that continuous functions are useful in approximating finite functions.

The panels now even provide your students with the means of differentiating and integrating single polynomial terms. One problem arises in the use of Panel 6.8.2. If the power of x, B, is an odd number, you will receive a decimal result, which may confuse them. You might ask them to integrate 4x⁶ with this panel to show that the result (although rounded) is what they expect. The panel result, 0.5714285714285714x⁷ is a bit unexpected. But you can have them check this: by integrating themselves they should get 4/7 x⁷ and they can check 4/7 against the decimal value by calculator.

Section 6.8 recalls the sequences that were met in Section 4.3 to introduce the idea of finite as opposed to continuous mathematics. Clearly this section does not try to do for finite mathematics even what the earlier sections do for calculus.

Of course, sequences play only a small role in this subject and you will probably wish to point out other topics that fall within this purview. The financial dealings of Chapter 4, for example, also fall within this branch of mathematics. As does much of probability.

To accent the extent of finite mathematics, ask your students to think of examples of finite and continuous mathematics. They will find that finite examples are much easier to identify.