Frequently Asked Questions

If I hear the same question from more than one person, I'll try to include the question and answer on this page as soon as I can. You can ask me questions before or after class, by email, or make an appointment to see me at another time.

Q: Problem 5 page 79 says to use the first exercise in section 1.4. Does this mean the first exercise in the text of section 1.4 or the first exercise at the end of the section?

A: Good point. Let's use exercise 1 page 58 (though you could use Exercise 1.4.9)

Q: I'm having trouble taking notes as fast as you are writing and at the same time trying to understand what you are saying.

Knowing what to write down as notes in a math class is a fine art that takes some practice. First be sure that you have read the assigned material before class so that you have some exposure to the material ahead of time. Some calculations don't need to be written down if they are simply algebraic manipulations that you are confident of doing on your own: For example, if I am solving a problem and end up with a quadratic equation, then use the quadratic formula to find the roots, you could skip the computation and just write down the roots. Keep in mind what you already have in the textbook. Don't just blindly copy down everything I write, but perhaps add to what is already in the text.
Some people have asked if it is OK to use a tape recorder, and that's fine with me.
Perhaps a better solution is to use the "groups" that were formed during the first quiz section. Have a designated "note-taker" and the rest in the group listen and try to understand. The "non-note-takers" could try to write down (during the lecture) just a few important ideas without the calculations. Any "ah-ha" insights they had during the lecture could be included. Then get together after class to xerox the notes, and compare comments.
Another suggestion is to ask questions. The amount of material that is in the text easily exceeds the amount I can present in lecture. So I have to make some guesses about what can be done on your own and how much I need to say about a technique. If I can get some feedback from you through questions, then I can tell what it is that you are understanding and what needs more explanation. Of course it is sometimes hard to formulate a question when seeing something new. Again it will help if you read the assigned material the night before. As several people have found out, if you ask me questions before class (I'm in the room at 8am) then I can incorporate the answers into my lecture. For example: Example 1.4.X doesn't make sense to me (perhaps the wording is ambiguous) or Why did this next formula follow from the above one (perhaps the author left out some steps) or I don't understand what this concept means (perhaps I can give more examples in class).

Q: Why does the method (to find a quadratic through 3 points) shown today (1/17) work?

A:Very good question. I'm impressed that someone was thoughtful enough to ask it in class.
- First let me give a summary of the method since it is not in the book. To find a quadratic through (x_1,y_1) (x_2,y_2) and (x_3,y_3) write down the following function:
  
  y=A(x-x_1)(x-x_2)+B(x-x_1)(x-x_3)+C(x-x_2)(x-x_3)
  
  Then plug in x=x_1 and y=y_1. Note that this will give an equation involving only C (not A or B), which is easy to solve. Do the same with the other two points, obtaining B and A.
- Why does it work?
  
  First note that you can solve for A, B, and C by the above method provided the points x_1, x_2 and x_3 are distinct. Then the above function will satisfy y(x_1)=y_1, y(x_2)=y_2 and y(x_3)=y_3. Finally notice that y is a quadratic function (simplify y).
- Remark: perhaps it is worthwhile to mention the answer to the same question for the method presented in the book. (Solving 3 simultaneous equations in 3 unknowns.) Here it is harder to prove that you can always solve for the coefficients of the quadratic. It depends on mathematics you have probably not seen (though is rather beautiful): If w is the column vector with entries (c b a), if u is the column vector with entries (y_1 y_2 y_3) and if V is the 3-by-3 Vandermonde matrix
  
  1 x_1 (x_1)^2
  1 x_2 (x_2)^2
  1 x_3 (x_3)^2
  
  Then the problem is equivalent to solving Vw=u for w. This is possible provided the determinant of the matrix V is not zero. This determinant equals
  
  (x_1-x_2)(x_1-x_3)(x_3-x_2)
  
  and is non-zero precisely when the points x_1, x_2 and x_3 are distinct.

Q: I'm stuck on part b of problem #4 page 91.

A: Hint: Use your answer to part a. to write the area in terms of just w.

Q: Can I just use the method I learned in high school to (find the equation of a line) (factor this polynomial)(divide a polynomial by another)(find a quadratic through 3 points)....?

A: You can use any method that is correct on any exam, so long as you show your work (we have to be able to understand what you write). However, the following may put this problem in perspective: Last year I noticed that my youngest son was having progressively more difficulty in math. After closely watching him solve some problems, I found the difficulty. He was using repeated addition to solve problems that were easier to do with multiplication. After questioning him about it, I found that he knew how to multiply but was not comfortable with it. He already knew that he could solve the problems by using the familiar, and comfortable, technique of repeated addition. So he resisted learning a new way to do something he could already do (and I might add that since this had been going on for some time, he had become quite adept at adding quickly!) In this course we try to show you techniques that we feel are the best in the long run. Our judgement is based not only on observing how students solve problems in this course, but on knowledge of the kind of problems you will encounter later in your mathematical career. The moral is: learn the technique we are teaching well enough to be comfortable with it, then decide whether to use it or another technique you are familiar with.

Q: I'm having trouble printing copies of the quiz solutions with Netscape. The copies are too big for the paper.

A: Be sure that you choose Portrait orientation and Letter Size (8-1/2 by 11) when the print menu comes up. The pictures are 8-1/2 inches wide on the math department machines. If someone could tell me how much smaller the pictures need to be, I'll try to reduce them. I tried cutting the size in half and they were not legible. Another way to view the files is to first save the image as a file (click on the image with mouse button 3 and you will be prompted to give the file a name). Then use a viewer such as xv to look at the file. e.g. if you save the file as quizsoln.jpg, then the command

xv quizsoln.jpg

will open a window then clicking mouse button 3 on the window will give you several options, such as reducing the size of the picture. If you would like higher resolution, or PostScript format let me know and I'll try to do that on the next quiz solution.

Q: I don't understand problem #10 page 228.

A: Hint: Carefully go over the problem about the engine piston in Friday's lecture. For part e, you may have a mess of an equation on one side, but if you bring the part that does not involve a square root over to the other side and then square the equation, then use the fact that sin^2(a) + cos^2(a)=1, it should simplify.

Q: In the lecture on 2-21-96 you were trying to find a minimum distance using the distance formula. Why is it the same as finding the minimum of the square of the distance?

A: We found the distance d(t) between a bug and a point using the distance formula. It turned out to be the square root of a quadratic function of time. We know how to find the minimum of a quadratic function, so we found the minimum of the square of the distance. The idea is that if d(t)^2 is as small as possible then d(t) is as small as possible. The two quantities are not the same, but they are as small as possible at the same time. Another way to look at it is: if A and B are two distances, then A < B if and only if A^2 < B^2.

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