Math 124 K - Autumn 2007

Instructor:

Dr. Matthew Conroy

Office hours and email

TA:

Sarah Gilles
sections KA, KB

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Exam Dates

Midterm 1:
Tuesday, October 23
Midterm 2:
Tuesday, November 20

Final Exam:
Saturday, December 8
1:30 PM - 4:20 PM
location: Kane 220

December 14, 2007

Here is a solution for the fifth writing problem. Please take a look at it some time (maybe next quarter, after you've got your graded writing problem back).

Here are some general comments that I thought of while grading this problem.

1. People seem to have almost completely forgotten the first derivative test, and why it is needed. With particular examples, it is not enough to simply say f'(x)=0 at this and that value of x, and so therefore there are this or that many extrema. Always keep in mind that f'(a)=0 does not imply an extremum at x=a: consider, for example, y=x³ at x=0.
2. When counting extrema, one needs to say something about extrema resulting from where f'(x) is undefined. Note that the curve y=f(x)=x^(2/3) has an extremum, yet f'(x) is never equal to zero. The same is true for inflection points and f''(x).
3. "Arguments" that involve equations like "0=0" are not logically valid without words to support them (and possibly not even then).
   Consider this argument to "prove" 1=2:
   1=2 (what I'm trying to prove)
   1(0) = 2(0) (multiply both sides by zero)
   0=0 (obviously true!)
   
   Therefore, 1=2.
   
   What is wrong with this argument?
   
4. If the algebra you're doing seems horrendous, it probably is not necessary (I do not torture). See if you could simplify before diving into a mess of algebra, or ask for ideas on how to avoid the mess.
5. People really liked throwing the Fundamental Theorem of Algebra around, so let's make sure we know what it says. It says this: every polynomial $p(x)$ of degree n, with complex coefficients, can be expressed in a factored form as
   p(x) = a (x- z₁) (x- z₂) ... ( x - z_n)
   where the z's are complex numbers, but they aren't necessarily distinct. From this, we can conclude that the equation p(x)=0 has exactly n roots, but some may be repeats. For example, the polynomial x²-2x+1 = (x-1)(x-1)=(x-1)², so we have two roots: 1 and 1. Since the roots are complex, and hence might not be real, we can apply this theorem to polynomials we encounter in Math 124 (or 125 or 126) to conclude that we will have at most n solutions to a degree n polynomial equation.
6. A lot of people were too ambitious, and made a lot of incorrect statements, like "if the function is a linear function over a quadratic one, then the function will have one extremum". This is not true (there are examples with 0, 1 and 2 extrema), and the problem didn't require you to make such fine distinctions. Be careful of trying to prove things you don't need to prove.

December 13, 2007

Here is the updated grade record, with final exam scores and course grades. These are your actual course grades that I will be reporting to the registrar (so they include the curve, dropped lowest homework score, etc.). If you notice any errors in the record, please report them to me.

I will be in my office on Friday from 1 to 2 pm if you would like to see your final exam or talk about any grading questions. You are also welcome to stop by any time next quarter.

Have a good break!

December 11, 2007

Here are the final exam stats: n=52, min=38, 1st quartile=82.5, median=92, 3rd quartile=97, max=100 (2 people).

December 10, 2007

Grading of the final exams is going swiftly. I expect to have the grade record updated here by Thursday evening. I also intend to have an office hour on Friday, from 1 to 2 PM, in Padelford C-544, if you wish to see your final exam. At the moment, I have a nasty head cold (or something), so there is a possibility I will cancel this office hour: just check here on Friday morning to see if I've added a cancellation message.

If I don't see you Friday, you can always stop by during my office hours next quarter to see your final exam. My hours will be listed here after the start of Winter quarter.

Have a good break!

December 5, 2007

Here is a nice website. It provides tools for graphing functions (and for doing other things). Enter the function you want to graph at the top, then select "The curve of f" and enter an interval of x values for the graph. Then press Show. Presto!

December 3, 2007

Writing Problem #5: Rational functions, sort of generally

Due this Friday, December 7. Since class is cancelled for this Friday, please drop by my office during my extended office hours on Friday to turn this in.

Consider a rational function p(x)/q(x) where p and q are polynomials with degree no more than 2. How many local extrema can this function have? How many inflection points can it have?

Be sure to not give just bounds (e.g., there are no more than 17 inflection points), but actually show which values are possible for the number of extrema and the number of inflection points.

December 1, 2007

Look at the discussion board (link at right) for suggestions for studying for the final exam and for links to old final exams.

November 30, 2007

Here are two applets to help you think about what we are doing in these optimization problems (like those in section 4.7).

This one illustrates the classic rectangular enclosure problem(s), like 5, 7, and 9 from section 4.7.

This one illustrates the problem of making a Norman window (see problem 28 from 4.7).

November 29, 2007

Here is a review sheet, or course summary, to help you prepare for the final exam.

November 28, 2007

Here is the grade record. Please check it right away to make sure your scores are recorded correctly.

Your grade information is listed under the last four digits of your student id number. Be sure to check that your scores have been recorded correctly. If you find an error, please bring it to the attention of your TA right away. Included is also an estimated course grade. This includes the course curve, and the dropping of your lowest homework score. Obviously it does not include the final exam: that can have a very large impact on your actual course grade.

November 27, 2007

Answers to the second midterm are here.

November 26, 2007

Here are some statistics for the second midterm:

n	52
1st quartile	59.5
median	64.5
3rd quartile	68
maximum	70 (6 people)

November 26, 2007

Please print this worksheet and bring it to section tomorrow. Thanks.

November 20, 2007

Here is the announcement of writing problem #4:

Writing Problem #4: How far is the horizon?

In this problem, you will develop an approximate formula for the distance to the horizon. We assume that the earth is a perfect sphere with radius r, and that the viewer's eye is a distance h above the surface of the earth. For the purposes of an approximation, we may assume that h is much smaller than r, say, no more than 1% of it.

1. Start by working out two exact expressions for the distance to the horizon along the surface of the earth, one involving arc tangent, and one involving arc cosine.
2. Use a linear approximation for arc tangent to make a simplified, approximate formula for the distance to the horizon; use the fact that h is much less than r to simplify it even further.
3. Explain why a linear approximation for arc cosine would not be useful. Instead, argue that cos x is approximately equal to 1-(1/2)x² for x near zero. Do this by comparing the functions and their first and second derivatives, concluding that this is the best possible quadratic approximation for cosine at zero. Then use this approximation to get an approximation for arccosine, and apply it to your formula to get a simplified, approximate formula for the distance to the horizon. Again use the fact that h is much less than r to simplify it even further.
4. Discuss the accuracy of your approximate formula(s). Feel free to include numerical and/or graphic evidence of the quality of the approximation. Using the actual radius of the earth for r, how large can h be while maintaining an error of less than 1%?

This problem is due November 28, 2007.

November 19, 2007

CLUE is holding a review session for Math 124 tonight at 6:30 in Mary Gates Hall, MGH 241.

November 15, 2007

Here is a review sheet for the second midterm.

November 7, 2007

Here is the related rates handout from Monday's and Wednesday's lectures.

November 5, 2007

Please print this worksheet for Tuesday's quiz section (you already have it if you have the coursepack).

October 31, 2007

Here is a solution to last week's writing problem. The key step in this solution is the observation that since the limit exists, and the limit of the denominator is zero, then the limit of the numerator must also be zero. There are a few different ways one can solve this problem, but this key step, in one form or another, needs to appear. We'll talk a bit about this problem in class on Friday.

October 30, 2007

Here is an applet illustrating this week's writing problem.

October 29, 2007

Here are some statistics on the first midterm exam:

n	52
1st quartile	56
median	62
3rd quartile	64
max	65 (10 people)

People did very well on this exam. It is difficult to give a guide to corresponding grades. I can break the scores into three groups:

• Those who scored above 60 did very well: such a score would translate to about 3.8 or above.
• Those who scored between 50 and 60 did well: such a score would translate to something between 2.7 and 3.7.
• Those who scored below 50 are on the low end of this class. Please see me as soon as possible to discuss how you can improve in this class.

Answers to the first midterm are now on the "old exam" archive, linked at right.

October 29, 2007

Writing Problem #3: due Monday, November 5

Consider a circle with large radius centered on the y-axis. Suppose its center is very far above the x-axis, so that it fits entirely "inside" the parabola y=x².

Then, move the center of the circle toward the x-axis, until the circle just intersects the curve y=x².

Now, reduce the radius of the circle while moving the center of the circle toward the x-axis so that the circle stays in contact with y=x².

Consider what happens as the radius is reduced to zero.

Describe the relationship between the height of the center of the circle and the radius of the circle (for all possible values of the radius).

October 28, 2007

Here is the worksheet for this Tuesday. Please print it and bring it to section.

October 19, 2007

CLUE will be holding a review session for Math 124 on Monday, October 22 from 6:30 to 8:00pm in Mary Gates Hall 241.

October 17, 2007

Take a look at this page for some thoughts on the wolves and caribou problem.

October 16, 2007

I've finished grading the first writing problem and I'll return them in lecture tomorrow. Most people did quite well on it. The most common issue was the fact that the equation |x²-1|=|y²-1| is equivalent to four equations without absolute value under certain conditions, and these conditions put restrictions on the values of x and y that make up the solution. It is important to consider these conditions when we eliminate absolute value from an equation.

Other general comments that will help in later writing:

• Using mathematical notation is okay! In fact, it is encouraged. Please don't write out things like "x squared minus one is equal to y squared minus 1" when you can just write x²-1=y²-1. Use words to convey things that cannot be conveyed with mathematical notation.
  
  Similarly, you don't need to include words of explanation for every little bit of algebra or arithmetic.
• You should pay attention to the distinction between equations and functions. For instance, the expression |x²-1|=|y²-1| is an equation, not a function. It may help to remember that a function is a relationship between two variables (call them y and x) so that there is at most one value of y for each value of x.
  
  An equation is any algebraic expression with an equals sign (=) in it.
  
  We often refer to a generic chunk of algebra as an "expression".
• We generally use the phrase "parametric equation" to refer to a situation where two quantities (often x and y) are related to a third quantity (often t) by two functions, e.g. x=f(t) and y=g(t). An equation relating x and y is not a parametric equation.
• Notice that the equations y=|x| and y = ± x are not equivalent. Notice that the first one defines a functional relationship between y and x (i.e., there is one value of y for each x) while the other does not. Think about what the graphs of each equation looks like (hint: they are different).
• The terms domain and range are used to refer to sets related to a given function. An equation (i.e., a non-function) does not have a domain or range, per se, so make sure you are saying what you want to say. You may have to resort to such language as "the set of y values for which this equation is true".
• Sometimes in mathematics, one can use a right-pointing arrow to stand for the word "implies" or "which implies". If you use it, be sure that you can grammatically substitute "implies" or "which implies" in its place.
• I don't know why, but a lot of people were throwig the word "essentially" around needlessly. If you write this word, consider whether leaving it out would change the meaning of what you've written. If not, leave it out. Saying "essentially" suggests that the situation is not exactly as described, and makes the reader expect some additional information to clear up the gap between what exactly is happening, and what "essentially" is happening.
• A sentence starts with a capital letter.

October 9, 2007

Check out this page with some interesting limit examples.

October 8, 2007

I'm holding extra office hours this week to facilitate the required office visitations. Here are my office hours this week, all in my office Padelford C-544:

Monday	10:30-12:15
Tuesday	2:30-3:30
Wednesday	10:30-12:15
Thursday	2:00-3:00
Friday	10:30-12:15, and 3:30-? (to whenever there is not a stream of students at my office door)

Check this page for help in finding my office.

October 7, 2007

I want to make sure that everyone is clear about what is expected of them with this first writing assignment.

The problem is to describe the graph of the equation |x²-1|=|y²-1|. This description should include supporting statements: that is, it should be a description together with an explanation of how you know that description to be true.

The idea here is to practice getting mathematical ideas expressed clearly in writing, using words and mathematical symbols. Here is something to try: imagine giving your description over the phone; every word you say into the phone should be written on the paper. Since we are writing mathematics, some of the words may be expressible using mathematical symbols. For instance, "<" can stand for the words "is less than".

Useful words and phrases include "thus", "so", "as a result of this", "because", "consider", "if", etc.

Take a look at examples 4 and 5 (pages 66 and 67) from Stewart to see the sort of writing that I expect. Do you see all those words? Do you see how they help you to understand what's going on? Do you see how they are integral to the expression of the mathematics? That's what we're shooting for.

October 2, 2007

Please bring this worksheet to quiz section Tuesday, October 9. It is in the course pack, or you can print it.

September 30, 2007

Speaking of parametric equations, you might like to have a look at the web page for a seminar I ran last spring. On the right of this page are a bunch of links to applets I wrote to demonstrate a variety of curves, many of which are defined parametrically.

Fun stuff, I think.

September 29, 2007

In this week's homework assignment, there are references to a coupld of "web supplements". These are available on the 124 Materials Website, and may be helpful in solving the homework problems. Here are some direct links:

• Sinusoidal Functions
• Tangent Lines
• Parametric Equations
• Exponential Modeling

September 28, 2007

The worksheet for next Tuesday is here. It is also included in the course pack. Please bring a copy of the worksheet to quiz section.

September 26, 2007

So, that seminar I was trying to announce today is listed on this page, in the seminars section. It is

H A&S 350 C: "Interfaces Between Mathematics and Industrial Applications"

Check it out, if it sounds like your kind of thing. It meets Fridays, 2:30-4:20.

September 23, 2007

Welcome to Math 124K, Autumn quarter 2007.

Announcements and other useful things will be posted here during the quarter, so check this site frequently.

Resources:

Homework Schedule

Course Syllabus (pdf)

Course Discussion Board

Dr. Conroy's old exams

Math 124 Materials Website

Other UW resources:

Math Study Center

Student Counseling Center

Information for Students of International TAs

Center for Learning
and Undergraduate
Enrichment (CLUE)