MT101 Fall 2003

Reading Assignments (and comments)

The reading assignments are from Calculus, Third Edition, by Hughes-Hallett et. al. and the comments refer to this. 

A remark on reading:  many students enter the university seeing a math textbook solely as a source of examples and homework.  They skip the exposition, look at the problems done in the text, and then try to do the homework.  This book does not provide pages of detailed mathematical exposition, but instead focusses on providing succint explanations that are truly useful.  I strongly encourage you to read the text as well as look at the examples and problems.  Even if you've already seen some of the material here (as have many in this class), the book can offer you new insights that are worth obtaining!

Week of:

September 3

Preface, page xi--To Students: How to Learn from this Book...the information given here is worth reading carefully!!

Section 5.1.

Comments:  As I mentioned in the remark above, you might be used to using your math book as follows: skip through the explanations and look at the sample problems, then try to do the homework. IT'S TIME TO TRY SOMETHING DIFFERENT. I want you to read the text carefully. This means working to understand the words of explanation, formulas and pictures as well as the sample problems.Employers report that an important skill they seek is the ability to read and understand technical language. If you do the reading of our text carefully, you will develop this. And you will find the lectures more comprehensible (as well, I hope, as enjoyable!), and the homework problems more useful.


September 8

Section 5.2, 5.3, 5.4.

Comments: You may have had Section 5.2 in a high-school class. It's worth reviewing this since later on we'll work on different methods for numerical integration and this section will be used. Section 5.3 captures an important aspect which will be a theme in our course: how is the mathematics used in the real world? All 4 sections (total change, units, averages, and applications to 2 moving objects) are important. Unit analysis is used frequently in science to be certain that the formulas are right (and when they're not, as with the NASA Mars probe which failed due to a failure to convert miles to kilometers, it really matters!). The two vehicle problems of examples 4-5 are good to go through carefully. If you understand them, then you will do well on the homework.Section 5.4 is a more formal mathematics section. We will not concentrate on the proof of the Fundamental Theorem, and the book isn't as clear as I'd like on the relationship between the argument on page 245 and the prior sections. So read this but don't worry about it. Focus on the remaining parts of the section. (The symmetry trick on page 246 is sometimes quite useful; the properties in Theorems 5.2 and 5.3, though formal, are ones we will use later, so it's worth learning them. Theorem 5.4 plays an important role in formal proofs, but will play a lesser role in our class.)


September 15

Section 6.1, 6.2

Comments: Section 6.1 emphasizes the graphical understanding of anti-derivatives. To understand this you need to remember that if f'(a)=0 then a is a critical point, and the graph of y=f(x) will have a horizontal tangent when x=a. Even if you've had antiderivatives in high school, you might well need some practice to go from the graph of f'(x) to the graph of an antiderivative f(x). Section 6.2 begins with some formulas for specific antiderivatives. We'll use them often, so you should learn them right away if they're not already familiar. Theorem 6.1 is a formal statement which you'll probably use without thinking about it much; notice that it looks a lot like Theorem 5.3. The last section, Using Antiderivatives to Compute Definite Integrals, is mportant; it is Theorem 5.1 again. This also can be confusing. Here's why: the DEFINITE integral (the one with upper and lower limits) is about AREA, while the INDEFINITE integral (the one without any limits) is a symbol saying take the ANTIDERIVATIVE. It is the Fundamental Theorem of Calculus (Theorem 5.1) which explains that these two different ideas are related. The reason we use similar notation (the integral sign) for both of them is that the FTC is true. The notation serves as a memory aid, but may also mask the deep fact that of this relation. Anyway, the end of Section 6.2 serves as a reminder of this and emphasizes that this is what is used to compute many definite integrals. (In the real world, this approach is used sometimes, and numerical methods--which we will work on soon--are also used extensively.)


September 22

Section 6.3 (The reading assignment is light this week due to the examination on Friday.)

Comments: We will work more with differential equations during the last month of our course. As you can see from this section, one source of differential equations is the problem of understanding motion. If a ball is thrown straight up or straight down, what happens? When does it hit the ground? How fast is it going? As you can imagine, similar ideas allow one to model balls thrown upward and outward, such as baseballs being thrown from left field to the plate; though this is not covered in the book, I will do one example in class. Historically, the mathematics of motion is of great importance. Indeed, fully understanding the motion of complicated systems involving more than one gravitational force is still an unsolved problem (of great contemporary interest). By the way, there are many differential equations which arise in economics, ecology, and other areas. We shall study some of them in Chapter 11.


September 29

Sections 6.4, 6.5, 7.1, 7.2

Comments: Section 6.4 is one many students struggle with. Please keep in mind the meaning of the DEFINITE integral as signed AREA. Thus Theorem 6.2 (The Second Fundamental Theorem of Calculus) is a statement about making an "area function" denoted F(x) on page 279, which counts the signed area under the graph y=f(t) between a and x. The theorem states that the DERIVATIVE of this "area function" is the function f(x). You will not be responsible for knowing the proof for this course (alas--I think it's worth knowing). It would be worth playing with your graphing calculator in reading example 1---try graphing the function sin(x)/x.

I am assigning Section 6.5 as reading, though we will probably not cover it in class. Anyone hoping to teach high school should certainly be familiar with this.

On to Chapter 7! Section 7.1 covers integration by substitution and may be familiar to many. The section is well-written, with well-chosen examples. To learn this method (either for the first time or as a review), you'll need to do lots of practice. There's simply no other way. Section 7.2 covers an important method called integration by parts (again, familar to many). I would guess that you'll find the boxed rules less useful than seeing and then doing a bunch of examples, but of course you're free to make use of them as you wish.


October 6

Sections 7.3,7.4,7.5

Section 7.3 is important. It covers using tables to compute integrals. This is the technique you're most likely to use in the years to come when you meet an integral in your work. To use the formulas you may need to use a preliminary substitution, as explained in the text. There's something surprising here: it's very easy to MIS-USE a formula from a table, forgetting to multiply by a constant or misunderstanding the formula. So please read the examples carefully and go over them. When I taught the course last year I was surprised by how many mistakes people made in applying the formulas.

Section 7.4 covers two techniques for doing integrals: partial fractions and trig substitutions. We'll only cover the first of these in our class (it will get used when we solve the logistic differential equation in Chapter 11). The trig substitutions are what is behind many of the formulas in the tables from Section 7.3, but will not be covered here. (You might want to read it for general information, especially if you're a math minor or planning on taking many quantitative courses.)

Section 7.5 is another important section. It is concerned with approximating integrals numerically. This is also frequently used in the real world, since (a) not every function has an antiderivative which is given by a simple formula and (b) sometimes you don't have a function, but just a bunch of data points which are points on the graph of the function, and you want to approximate a definite integral from your data points. The topics of this section are guaranteed to turn up on the next midterm.

The various methods for approximating an integral are implemented by program INTEGRAL, which you should load onto your calculator and practice using.

October 13

Section 7.6, 7.7

Section 7.6 is important for the same reasons that section 7.5 is. It introduces Simpson's rule, and also gives an answer to the question of how good the various approximation methods are--how much extra work do you need to do (with your calculator) to get an extra decimal place of accuracy in your approximation? As you can imagine, this is also important in the real world, where you want to approximate an integral but not have approximation errors accumulate too much. Last year students found this section difficult, so I recommend that you plan to read it several times. Notice that the information about errors isn't proved but is only stated in a rough way based on a numerical example. I'll state more precise formulations in class, though I won't give any proofs.

Section 7.7 concerns improper integrals, either integrals "out to infinity" or integrals where the function you're integrating isn't well behaved somewhere in the interval you're integrating over. This is useful since sometimes you're integrating to infinity for all practical concerns, as in the application to Energy given on page 329. There are many different types of improper integrals. To keep them straight, please go over the examples in this Section carefully.


October 20

Section 7.7, 8.1.

Please finish reading Section 7.7 if you have not already done so. Coments are given in last week's reading assignment.

We will not cover Section 7.8 and you do not need to read it.

In the next unit of our course, which we will begin in earnest after the examination, we will take up applications of the definite integral. Section 8.1 gives applications to areas and volumes. (Though you may have had applications of the integral to solids of revolution in a previous course, the material here is different, and you do not need to know anything about solids of revolution.) Besides Calculus, the main tools are high school geometry, in particular similar triangles and the Pythagorean theorem. If you are rusty on these topics please look at the examples carefully, and don't hesitate to come see me for extra review.

Due to the examination this coming Monday, I won't assign further reading from Chapter 8, but this will make some future reading assignments concurrent with the presentation of the material in class. So I urge you to do these reading assignments early in the week.

October 27

Sections 8.2, pages 356-357 only, 8.5.

Due to the examination on Monday, there is a short reading assignment this week. In Section 8.2, we will only cover the arc length of a curve. There's a formula here, and that's fine, but I want to point out that this section is important for another reason. Namely, it's valuable to see where the formula comes from--subdivide, approximate the arc length by a sum of lengths of line segments, and let the number of subdivisions go to infinity. Then the sum becomes a definite integral. The point is that there are lots of other problems in a variety of fields where a similar method leads you to an integral. We'll see some later, but it's easy to get caught up in memorizing formulas and thinking that this is the main thing. It's not. If you understand where these formulas come from, you'll be much better equipped to use these ideas in your own subject later in your career. We'll cover page 356. Page 357 (arc length of a parametric curve) is worth reading once but won't be discussed in class, and you're not responsible for it on the examinations.

Section 8.5 concerns applications of the definite integral to economics. We'll focus on present and future value and on income streams. If you're an economics or finance major, you should read the sections on supply and demand curves and on consumer and producer surplus, where various areas with economic meaning are written as integrals involving the supply and demand curves. For our course you are only responsible for the material on pages 377-380 (through Example 2, page 380).

We will begin Section 8.6 on Monday, so you would be well-served to read this over the weekend of November 1-2.

November 3

Sections 8.6, 8.7

These are important sections and cover materials related to statistics. The question behind them is how to describe data, and this is of vital importance in many areas of the natural and social sciences. (You can even imagine it being used in the humanities, for example to describe the range of words used in Shakespeare's plays.) The book does a very good job with these sections, though the top of page 393 may require a slow and careful reading to understand.


November 10

Sections 11.1, 11.2, 11.3

We will spend the rest of the semester working on differential equations, Chapter 11, Sections 11.1 to 11.7. These sections are well-written and make useful reading. As with integration, we will be concerned with solving differential equations numerically, graphically, and exactly. In this course we shall focus on one specific type of differential equation, a first order differential equation.

Section 11.1 introduces differential equations, explains what "solving" a differential equation means, and suggests how we might solve one specific equation numerically. Please go over the discussion on page 478 with care. In Section 11.2 we learn about solving differential equations graphically through the use of slope fields. I will supplement the examples in the text by giving additional examples in class. Section 11.3 is concerned with the numerical solution of equations via Euler's method. Once again we shall approximate a curve (the solution curve) by a collection of line segments.

November 17

Section 11.4.

In this Section we complete our analysis of abstract first order differential equations by finding a way to solve certain equations analytically--that is, to generate a formula for the solutions. Please be sure to go over the examples in the text. There is a short reading assignment this week due to the examination.  

December 1

Sections 11.5, 11.6, 11.7

In the last 4 lectures of our course we shall be concerned with applying our knowledge of differential equations to real-world phenomena. In Section 11.5 we examine growth and decay situations, where the change is proportional to how much is there. (We mentioned this briefly when we discussed continuously compounded interest in Section 8.5.) The example with Newton's law of heating and cooling is especially important as it leads to a discussion of equilibrium solutions. That is, different solutions to a differential equation may have very different characteristics, and (as we shall see in class) it is important to understand these differences. The notions of equilibrium solution and stable vs. unstable equilibrium are very important.

Section 11.6 gives some specific applications of differential equations to modelling. We will cover the net worth of a company in class. Please read the entire section, though. In the final section of text for our course, 11.7, we shall study models for population growth, and in particular formulate the logistic model. We shall study its solutions both qualitatively (pgs. 518-519) and analytically (pgs. 519-520). (Note the use of partial fractions in the analytic solution on page 519, the sixth /seventh lines of the section "The Analytic Solution to the Logistic Equation.") The excellent exercises for this section illustrate the many ways that these ideas can be applied to real-world problems.

Last revised:  July 21, 2003.  Copyright 2003 by Solomon Friedberg. All rights reserved.