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Have a great break.
| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| Review | Problems from 121 exams | None | |
| 5.1 | #1-29 odd, 45,47 | #22,44 | Wednesday, September 7 |
| Flow rate | #1-3 from handout and plots | None | |
| 5.2 | #1-11 odd, 17,19,23,25 | #16,24 | Friday, September 9 |
| 5.3 | #3,7,17,19,27,38,39 | #26,33 | Monday, September 12 |
| 5.4 | #1-21 odd, 25,53,55 | None | |
| 5.4 | #39,41,43,57,61 | None | |
| 5.5 | #1-43 odd | #20,40 | Wednesday, September 21 |
| 5.7 | #3,7,19,23,27,31 | #28,32 | Friday, September 23 |
| 5.8 | #3,5,7,11,13,17,21,27,31 | See daily note | Wednesday, September 28 |
| 6.1 | #1,3,5,9,11,13,15,19,25,27 | #18,24 | Thursday, September 29 |
| 6.2 | #1,3,5,9,15,19,37,39,51,55 | #40,56 | Monday, October 3 |
| 6.4 | #5,15,27,33,37 | #16,28 | Thursday, October 6 |
| 6.3 | #7,9,11,17,21,23,25,39,43,45,49 | #42 | Monday, October 10 |
| Constructing definite integrals | #1,2,3 from handout | None | |
| 7.1 | #1-41 odd | #18,40 | Thursday, October 20 |
| 7.2 | #1-27 odd | #14,26 | Friday, October 21 |
| 7.3 | #5,7,15,17,25,35,37,39,43 | #8,42 | Monday, October 24 |
| 7.4 | #1,3,5,15,17,19,23,17 | #18,22 | Wednesday, October 26 |
| 7.5 | #1-71 odd | None | |
| 7.7 | #1-41 odd, 47, 49 | #24,48 | Monday, October 31 |
| 8.1 | #3,11,13,17,19,25,27,31,37 | #28,42 | Friday, November 4 |
| 8.2 | #1,2,3-13 odd,17,25,47,51,57, (read 58,59) | #18,48 | Wednesday, November 9 |
| 8.3 | #19-41 odd, 55,58,62 | None | Wednesday, November 9 |
| 8.4 | #3,17,19,21,27,33,43,51,55 | #20,48 | Wednesday, November 16 |
| 8.5 | #9,13,15,17,19,21,27,31,33,41,43,45,49,53 | #26,34 | Thursday, November 17 |
| 8.6 | #9,11,17,25,39,43 | #22,40 | Monday, November 21 |
| Function approximation | #1-5 from handout | #2,4 | Wednesday, November 30 |
| 8.8 | #3,7,9,13,15,17,21,23,27,31,37,45,53 | #24,34 | Thursday, December 1 |
| Approximations and errors | #1-4 from handout | None | |
| 8.7 | #1,5,13,15,21,23,35,39,47,53 | #12,26 | Monday, December 5 |
The Mathematical Atlas describes the many fields and subfields of mathematics. The site has numerous links to other interesting and useful sites about mathematics.
If you are interested in the history of mathematics, a good place to start is the History of Mathematics page at the University of St. Andrews, Scotland.
Check out the Astronomy Picture of the Day.
Your focus for now should be recalling all the things you've learned about first-sememster calculus topics. You should look through Chapters 1 through 4 of the textbook to see how limits, continuity, and derivatives are covered. You should also look at the exams from the last time I taught our MATH 121 course. You should work on problems from the final exam. Bring questions you have on any of the exam problems to class on Wednesday. You can skip over problems that deal with integrals and integration.
There are two supplements to the text that you might find useful. One is the Student Mathematics Handbook and Integral Table for Calculus. This reviews topics from geometry, algebra, and trigonometry. It can serve as a useful reference. The Student Mathematics Handbook is included on a CD that comes with the text when it is purchased new. You might want to check if you already have this. The other is the Student Survival and Solutions Manual. This has solutions for the odd-numbered problems in the text.
Understanding limits deeply requires understanding something about the structure of the real numbers. One way to think about real numbers is in terms of a number line: Each real number corresponds to a point on the number line and each point on the number line corresponds to a real number. There are no gaps or holes in the number line. The number line is also "infinitely stretchable". A piece of the number line, such as the interval between 0 and 1, can be stretched out to look just like the number line itself.
At times, it is useful to think of the real numbers as composed of rational numbers and irrational numbers. A number is rational if it can be expressed as a ratio of two integers. A number is irrational if it cannot be expressed as a ratio of two integers. It's usually hard to prove that a given number is irrational, but there are lots of irrational numbers. It is a fact that if n is a not a perfect square, then the square root of n is irrational.
Many people think about the difference between rational numbers and irrational numbers in terms of decimal expansions. The decimal expansion of a rational number will end in a repeating block of finite size. (The repeated block may contain just the digit 0 which we generally don't write explicitly.) The decimal expansion of an irrational number will not end in a repeating block of finite size. The number 0.101001000100001..... is irrational. The decimal expansion of this number has a pattern but that pattern does not end in a repeating block of finite size.
Here's a fundamental fact about the real numbers: Between any two real numbers, there is at least one rational number and at least one irrational number. These leads to lots of interesting consequences including the fact that between any two real numbers there are infinitely many rational numbers and infinitely many irrational numbers.
So after all that, here's a possible response to Problem 7 from Exam 1: "The only two outputs of the function are -1 and 1 so these are the only possible limits. The number -1 cannot be the limit because there are outputs f(x) that are not near -1 for inputs that we can take as close to 5 as we want. To get an output that is not close to -1, we just use a rational number as close to 5 as we want. The output for that rational number is 1 which is not close to -1. For a similar reason, the number 1 cannot be the limit because we can get -1 as an output for inputs as close to 5 as we want by using irrational numbers. Thus, this limit does not exist."
You might be feeling a bit rusty on some computational skills. It might be
useful to look at some examples and do some problems from the first part of
the text. There are answers to the odd numbered problems in the back of
the text. Here's the sections you might find most immediately useful:
Section 2.2 on computing limits
Section 3.2 on computing derivatives
Section 3.3 on derivatives of trigonometric functions
Section 3.5 on the chain rule
Sectoin 4.5 on L'Hopital's rule
Of these, Section 2.2 is probably the lowest priority. It will be more
important to remember details of how to compute derivatives.
In class, you started working on the handout Flow rates and accumulations. You should keep working on this as homework. A key idea is that for a constant rate, accumulation is the rate multiplied by the time elapsed. For most of the five-minute intervals in the problems, the rate is not constant. You can estimate the total accumulation by using a constant rate for small time intervals. There are many reasonable ways to get an estimate. Any simple method is sufficient for this context.
As you work on this, think about the relationship between rate of change (in volume for this situation) and accumulation.
The usual motivation for defining definite integral is to look at the problem of computing area. The textbook does this in Section 5.2. In class, I am using the problem of computing accumulation to motivate the definition of definite integral.
The directions in the problem sets for Sections 5.2 and 5.3 refer to a formula for Sn. You should ignore this formula. It is much better to construct a Riemann sum by thinking through the definition.
Our first exam will be on Thursday, September 15 from 9:30 to 10:50. It will cover ideas from the review material and Sections 5.1 through 5.4 with focus on the latter. I will post a set of specific objectives early next week.
Our first exam will be on Thursday, September 15 from 9:30 to 10:50. It will cover ideas from the review material and Sections 5.1 through 5.4 with focus on the latter. For this exam, a well-prepared student should be able to
Here's an exam from a previous course that covers roughly the same material that we'll have on tomorrow's exam. Here's the standard disclaimer I give with old exams: Don't assume I am going to write our exam by just making small changes to the old exam. You can use this old exam to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.
I will check e-mail occasionally this evening and will respond to any questions you send me.
In discussing area today in class, I tried to emphasize a distinction between region and area. This distinction is blurred in our everyday uses of these words. In mathematical discussions, region refers to a set of points (a set of points in the plane for our purposes today). Area is a number that we associate with a given region. For example, consider the region to be a disk. There are several numbers we can associate with (or measure for) a disk: the radius, the diameter, the circumference, the area. Each of these quantities is a number that we associate with the region that is the disk.
If you were provoked by our discussion of area today, you might enjoy reading about the Banach-Tarski paradox. This link gives a very brief description. You can find many more by doing "Banach-Tarski" in a web search.
I've assigned a few problems on numerical approximations of definite integrals from Section 5.8. I'll assign more problems to work on after tomorrow's class.
For Section 5.8, the problem to submit is the following: Approximate the definite integral of e-x2 for the interval [0,1] with error less than 0.001.
To motivate Simpson's rule as a weighted average, here's a handout to compare the errors in the midpoint and trapezoid approximations.
I have forgotten to mention in class that we are skipping over Section 5.6 for now. We will return to this later in the semester.
In class, we talked briefly about Archimedes and how we know about his methods. You can read more about the "lost and found" manuscript on the web pages for The Walters Museum (where the manuscript is now being studied) and on the web pages for NOVA (which did an episode on this story). If you want to see the details of the way Archimedes got a formula for the volume of a sphere, you can read this presentation (as a PDF file) written by Doug Faires (Youngstown State University).
The main theme in Chapter 6 is how to build a definite integral to compute a quantity of interest. Sections 6.1 through 6.4 focus on geometric quantities such as length, area, surface area, and volume. Sections 6.5 and 6.6 focus on quantities from physics, finance, and other areas.
Section 6.3 introduces polar coordinates and polar curves. I've assigned problems on these topics. After mastering these ideas, we'll look at computing lengths, areas, and volumes involving polar curves. I'll assign more problems from Section 6.3 later.
The problems on the handout "Constructing definite integrals" give you practice in constructing a definite integral to compute a desired quantity. These problems also give you experience working with parameters rather than specific numerical values. This is typical of what you will see in using calculus as a tool in other fields such as physics, chemistry, biology, geology, economics, and finance.
Exam #2 will be Thursday, October 12 from 9:30-10:50. Check your schedule to see if you are free for this 80 minute block. The exam will cover everything since the first exam including topics from the end of Chapter 5 and most of Chapter 6.
When working on constructing definite integrals in class today, I saw lots of people jumping to an incorrect result without thinking carefully about the intermediate details. Below is an outline of what one often does in constructing a definite integral. The outline is not a recipe that one can follow in every case without thinking. Instead, it represents some concrete steps you can follow to put yourself in position to construct the correct definite integral.
Our second exam will be on Thursday, October 12 from 9:30 to 10:50. It will cover ideas Sections 5.5, 5.7, 5.8, and Chapter 6. For this exam, a well-prepared student should be able to
Our next major topic is techniques of antidifferentiation. It is important to master some techniques for getting antiderivatives "by hand" and to learn how to use tools such as tables of integrals and symbolic computing software. One tool available on the internet is The Integrator web site provided by a company called Wolfram. This software company produces the program Mathematica, a general purpose mathematical program that can do symbolic and numerical calculations and produce graphics of all sorts. I'll demonstrate some of the capabilities in class some time this semester.
We've now looked at all of the major techniques of integration/antidifferentiation that we will cover. These fall into three categories:
The problems in Section 7.5 give you an opportunity to practice antidifferentation outside the context of a section covering a specific technique. You should do as many of these problems as you think you need. I suggest you do attempt a few problems every day between now and our next exam (scheduled for Thursday, November 10). Try to view each problem as a little puzzle. Often, you can look at the puzzle in the right way and it's simple. We'll take a bit of class time each day in the next few weeks to look at questions that you have from these problems.
Our topic of study now is improper integrals. There are deep and interesting conceptual ideas here. Part of what you will need to do is develop intuition for the nature of various functions. You will need to have a good feel for a variety of functions so that you can effectively make comparison arguments to determine whether a given improper integral converges or diverges. The same type of comparison argument will be essential when we study infinite series.
We are now turning our attention to sequences and series. Here are some informal definitions:
Exam #3 is scheduled for Thursday, November 10 from 9:30-10:50 am. It will cover techniques of integration (the material in Sections 7.1-7.5) and the concepts of improper integrals, sequences, and series. Calculators will not be allowed for the section of the exam dealing with techniques of integration.
Given a series, the first question to ask is Does the series converge or diverge? In addressing this question for any particular series, I will expect you to provide an argument supporting your conclusion. We will develop a variety of standard arguments that you can use. In Section 8.3, two of these argument types are given. The first is the divergence test: if the sequence of terms in the series does not converge to 0, then the series diverges. The second is what I will call an improper integral comparision argument. The idea here is that, under certain conditions, the behavior of a particular series corresponds to the nature of a related improper integral. To use this argument, you will need to verify the conditions hold, identify the related improper integral, and determine the nature of that improper integral. Note: Most texts, including ours, call this the integral test. I prefer to think about this as a type of argument to be made rather than a test to be applied.
Exam #3 is on Thursday, November 10 from 9:30-10:50. It will cover material from Sections 7.1-7.5, 7.7, and 8.1-8.3. For this exam, a well-prepared student should be able to
We are continuing our study of series. Today, we talked about how to make direct comparison arguments and limit comparison argumentss. For these arguments, you need to have a conjecture on whether the given series converges or diverges and then bring in an appropriate comparison series.
Here's an answer to the question I posed in class.
We are now looking at the idea of approximating a given function by a polynomial function. In many situations, it is useful to trade in an exact but hard to compute value for an approximate but easy to compute value. Here, we will trade in a general function for a polynomial function. Evaluating a polynomial function uses only simple arithmetic.
Our basic idea in constructing a polynomial approximation is to notice that the linear or tangent line approximation is a first-degree polynomial approximation. With this idea in mind, it is not a stretch to think that a second-degree polynomial approximation will involve the second derivative of the original function, a third-degree polynomial approximation will involve the third derivative of the function, and so on.
The approximating polynomials that we will use are called Taylor polynomials. After we get comfortable computing Taylor polynomials of degree n, we will ask "What do we get in the limit as n increases without bound?" This limit is a series that we will call a Taylor series. This type of series is a little different than those we have dealt with so far because each term contains a variable. Taylor series are a special type of power series so we will soon study power series.
The text develops this material by first discussing power series (in Section 8.7) and then developing the ideas of Taylor polynomials and Taylor series (in Section 8.8).
Here is the handout looking at the error in approximating a function by a Taylor polynomial. The main idea we discussed today is finding an "easy-to-compute" upper bound on the error (or remainder term) so that we can determine how good a specific approximation is.
I've posted a handout with a few problems that involve determining an upper bound on the error in using a Taylor polynomial approximation. See the list of homework assignments below.
I've turned on a "score check" feature. Look for this in the menu at the top of this page. You can check that I have correctly recorded your scores for homework assignments, projects, and exams. Let me know if my records do not match your records.
Exam #4 is on Wednesday, December 7 from 10:00-10:50. It will cover material from Sections 8.4-8.8. For this exam, a well-prepared student should be able to