(Perhaps not so) Frequently Asked Questions
from the class of fall 2006
CALCULATORS
1. Will I need a calculator?
It is highly recommended for some of the class work, especially graphing functions and doing numerical experiments. I want you to learn to use mathematical software - Maple will be supported in this class - that is available in Clapp 401, 420 & 422 during Help Session times, Sunday through Thursday nights.
2. What kind of calculator should I have?
Any kind of "graphing" calculator; your instructor will be able to assist/support your use of a TI-83 plus or TI-84. A calculator, like the TI-89, which has symbolic algebra capability, will not be allowed for in-class test use.
3. Where can I get a calculator?
I have a supply from which you may borrow.
HOW IS THIS CLASS DIFFERENT? WHAT PREPARATION NEEDED? HOW TO REVIEW?
4. Are all of our classes going to be mainly discussion, or as we delve into concepts will it shift more to lecturing?
We'll have some pairs and small group work, but mostly it will be a sort of "interactive lecture" mode in which i will talk. Students should make comments and ask questions at any time - within reason! The interactive mode can turn into discussion at any time that enough students want to get more actively involved.
5. I took AP Calculus, the AB section. Where does this put me with regard to what I should know for Calculus II?
In principle, you should be prepared for Calculus II here at MHC. Most of Calculus II should be new to you, and you have the prerequisite background to understand these new ideas. Of course, in practice, none of us really learns everything we're exposed to, so there's often some "review," or what I think of as "learning in more depth" needed.
6. It has been a long time since I did any math. Do you have any sugestions for reviewing derivatives and integrals?
Take a look at <http://www.mtholyoke.edu/courses/jmorrow/calculus_ii/deriv_integ_solns.pdf>, which is linked from the home page. Try the Step-By-Step Differentiation link in <http://www.mtholyoke.edu/courses/jmorrow/calculus_ii/webcalc_resources.pdf> for more.
7. You said this class would be different from a high school calculus class in some ways. Could you elaborate on that statement.
The two may not be so different, depending on who the teacher is. Generally, though, the high school class focuses on procedural understanding, while a college-level class (my classes, anyway) focuses more on conceptual understanding, reasoning, non-routine problem solving, and abstraction. AP courses have to go fast and often don't go very deeply in some sense.
CLASSROOM POLICIES
8. I am curious about classroom policies a bit; is it worthwhile to bring say a laptop to class? Also, are water bottles allowed?
The laptop will probably be in your way unless you can take "mathematical" notes easily with it. You're certainly welcome to try. Water and food are allowed, but not in addition to a laptop!
HELP!!
9. Where can I get help for this course?
There are several sources of help:
STUDYING
10. I do all my homework, so why am I getting a C- on the tests?
Although there isn't a ready answer to this question, here are some ideas of things to do:
Talk to me about your study and homework methods
Rather than considering that the objective of homeworkto be "doing the problems," think of what the "problems are doing," what the problem solutions are illustrating and what other problems might be done using the same method of solution; make up problems for yourself. Your instructor will be happy to look over such additional "made-up" problems.
11.
What methods of studying do you recommend to keep up with the course
material and
to
learn the
material
most thoroughly?
I think that studying methods are pretty individual, but I would stress:
a. Reflection on problems you have solved - which problems used which
concepts/theorems/techniques? can I construct problems similar enough
to a problem in the book that I could use the same concepts/theorems/techniques
to solve my newly constructed problem?
b. Reading the text actively with pencil and paper at the ready to
check on/illustrate ideas and note what you don't follow in as detailed
a way
as you can.
c. Talking about what you are studying with people at different levels
and with different experiences, your classmates, me, your family,
your room-mate and friends.
SEQUENCES
12. SEQUENCE CONVERGENCE Although I have read and understood what convergent and divergent sequences are, especially from the graphs, how exactly do you decide whether or not it can fall under the category of divergence or convergence?
The question is totally in line with what a curious person (a mathematician) wants to know. And this is what most of section 11.1 is all about. Pages 704-710 answer the question to the extent it can be answered. The text has red boxes around all the principles you need to use. The rest of the reading explains why those principles are valid and illustrates (see Examples) their use. All of this is critical to answering the question.
13. FORMULAS FOR SEQUENCES Is there any easy way to find an explicit formula for a(n), the nth term of a sequence, when the first few terms are given without having to guess and look for patterns?
I think it's safe to say that you must look for patterns! It helps me to first look for a recursion formula by seeing if there is a pattern to the way that each term in the sequence is related to its immediate predecessor. If I can get a recursion formula, then I can often easily get an explicit formula in terms of some function of n.
14. SEQUENCE CONVERGENCE How can we prove mathematically that a sequence is convergent or divergent?
One way is to breaak the proof up into steps, with each step justified by citing a theorem from the textbook. I illustrated this method at Example1.
A more direct method: If you want to show that a[n] --> L as n --> infinity, you need to show that you can make |a[n] - L| as small as you want by choosing n sufficiently large. So, for example, to start the proof that 1-(1/n) -->1, you need to think about how large you would need to make n in order to make | 1-(1/n) - 1| "small," arbitrarily small that is.
15. SEQUENCE CONVERGENCE Are sequences of indeterminate 0/0 or infinity/infinity form always convergent?
No; look for a sequence where the numerator approaches infinity much faster than the denominator approaches infinity.
16. SEQUENCE CONVERGENCE Does L'Hospital's Rule always succeed in determining the limit of a sequence of the indeterminate 0/0 or infinity/infinity form? It doesn't seem to work on the sequence whose nth term is 2^(n)/3^(n+1).
No, the rule doesn't always work, as you suggest. For the example 2^(n)/3^(n+1), try rewriting the nth term as (1/3)(2/3)^n.
17. SEQUENCE CONVERGENCE When all other methods fail, can you solve a limit of a sequence problem by substituting large values of n into the formula for the nth term?
a. It's never a bad idea to look at the terms of a sequence for large values of n; the specific values you see may give you some insight into the nature of the sequence and help you formulate reasoning about the infinitely many terms you can't see. The specific numbers may also give you more confidence - or sometimes less confidence - in the more abstract reasoning you do about the sequence. The specific numbers may convince you of something about the sequence, although even if you're conced of the truth of something, the numbers themselves don't provide understanding of why something is true.
b. There are sequences which we can prove converge to a limit L, but we have no methods to determine the exact value of L. An approximate value for the limit is then a solution of sorts, especially if we can say something like, " The limit is approximately 4.371, with error no larger than 0.0005."
18. SEQUENCE CONVERGENCE Are there unbounded sequences that converge?
No; if a sequence a[n] is unbounded it can't get arbitrarily close to any specific number. It might, however, be the case that a[n] --> infinity or that a[n] --> negative infinity.
SEQUENCES VS SERIES
19. What is the difference between a sequence and a series?
Ah! You've found the point of most initial difficulty concerning infinite series. At the point in the course where we have not yet talked about infinite series, but only aboput sequences, I'll give a preview. An infinite series is written in such a way as to look like a sum of infinitely many terms. Since we can't really add infinitely many terms by ordinary addition, we "work around" that by creating a sequence. This sequence is called the sequence of partial sums. The sequence of partial sums has 1st term that is the same as the 1st term of the sum (series), 2nd term that is the sum of the first 2 terms of the series, 3rd term that is the sum of the first3 terms of the series, and so on. See p. 714 of Stewart.
SERIES CONVERGENCE
20.Theorem 6 says that if an infinite series is convergent, then its nth term must have a limit of 0; this means that if the limit of the nth term of a series is not 0, then the series is divergent. So how exactly can we know that a series is convergent when the limit of the nth term of a series is 0?
If the sequence of terms of an infinite series does approach 0, then
a. thinking of what you have done as a test of divergence of the series, the test is inconclusive
b. there is still a chance that the series could converge
c. you need to try something else!
21. Can you explain how one tells whether a series is convergent?
The text explains the meaning of series convergence on page 714, but we most often don't have to resort to this definition just to know whether a specific series converges. The author,however, uses over 40 pages in 6 sections of Chapter 11 to answer your question! The methods described in these 6 sections were invented by very clever people over a considerable period of time and represent a great human achievement.
PRECALCULUS
22. How does my previous study of functions and graphs relate to what we are doing with sequences?
There is a direct relation between the limit of a function as the real variable x approaches infinity and the limit of a sequence as the index n approaches infinity. The text tell us a theorem, numbered 3 on page 704, which relates/equates the two limits. As you look at the graph of a function (or sequence) from left to right and you look at the points on the graph (x,f(x)) (or (n,a[n])), the 2nd coordinates of those points will approach the limit that we're talking about if there is such a limit. You will also be able, many times, to see that the limit does not exist.
x. What's a pyramid exam?
A pyramid exam has 3 parts:
As a result, there is a pyramid of papers submitted: 1 individually-done paper per student, 1 team-done paper per team, and 1 class-done paper per class.
y. Why have pyramid exams?
Propose your own reasons! More to come later.