Homework Helpline

After checking to make sure that your question has not already been asked (and
perhaps answered), please submit your question on this form. Also, please note:
  1. This service is for help between classes with specific (as opposed to hopelessly vague) academic difficulties that arise between classes. It is not for complaints (which should be addressed to me either by electronic mail or, better yet, in person); and it does not excuse you from asking questions in class about matters that arise during class.
  2. This service is exclusively for the benefit of my current students at Florida State University. Your question will be answered only if you reveal your name to me on the submission form; however, your identity will otherwise remain a secret (see below for examples).
The Homework Helpline is no longer active.
It was decommissioned in October, 2001 because nobody was using it.

Saturday, September 15, 2001 at 23:27:13 (EDT)
MAC 2313
I was working some problems today and I ran across a "problem" with problem # 15 in chapter 12.2. I just can't fiqure out how to begin with drawing the vectors for this problem.
I have a problem with this problem too. Click here and here and here for details
Sunday, October 29, 2000 at 18:51:07 (EST)
MAC 2312
isn't the Taylor series of tan(x) infinite for every nonzero value of x, meaning that you can get an infinite amount of values that correspond to x, x^2, x^3...?
No, its radius of convergence (about the origin) is p ÷ 2.
Sunday, October 29, 2000 at 15:45:04 (EST)
MAC 2312
for question 2 on the test, what is a right circular cone?
The volume of revolution you get when you rotate a right-angled triangle through angle2p about a side other than the hypotenuse. For a diagram, see p. 377 of your text.
Wednesday, October 25, 2000 at 15:46:54 (EDT)
MAC 2312
For the mock test, I don't understand the reasoning on the limits of Integration for finding the mass; why did you choose -1 and 1 instead of maybe 0 and 2? Because it saved me the trouble of rewriting the density function. I think if I understand this, I can get the rest of the problem. And oh yeah, How does the product rule function when integrating? It doesn't. You have to integrate by parts instead. I don't remember. Thanks
Click here and here for details
Monday, October 16, 2000 at 20:09:41 (EDT)
MAC 2312
For #4(e), I found the equation for surface area in the book: 2pi*integral from a to b of(f(x) *(sqrt(1+(dy/dx)^2))). Plugging in the numbers is easy, but I end up with a polynomial fraction times a sqrt of another polynomial fraction. Is there a substitution that would help that I can't see? Or is there any other suggestions you could give on this problem?
If you can deal with the square root for #4(b), then you can deal with the square root for #4(e).
Monday, October 16, 2000 at 11:21:11 (EDT)
MAC 2312
When we turn in our assignment tomorrow, do you want it to be in blue or black ink, or is pencil acceptable?
It doesn't HAVE to be in pen; however, I would prefer it to be in pen. (The main rationale for insisting on pen in a classroom test is that it saves you all the time you otherwise spend erasing unnecessarily.) Moreover, regardless of whether you honor my preference or ignore it, your assignment has to be clear and neat (which, needless to say, does not in any way mean that you can't cross things out—only that you have to cross out neatly, so that the intended meaning is perfectly clear).
Monday, October 16, 2000 at 10:08:14 (EDT)
MAC 2312
Some quick questions: For #1: using washers, I got 4 pi * the integral from 0 to pi/2 of sqrt(a^2-cos (theta)) * a cos (theta) d(theta): How can I get rid of the a cos (theta)? Should I do another substitution, if so, what can I use? No. The problem (if you have made no mistakes) is to rewritecos(q)^2 in terms of something you know how to integrate. Take another look at the trigonometric formulae given on the mock test. For #2c: I got the correct answer (checked on my calculator), but it's negative.Then it's not correct. Are A=1, B=1, sideways fish= (1+sqrt(2)), and beta= (1-sqrt(2))? I have already said that A = 0. Please see my October 8 response. For #3d: I got the integral from 4 to 9 of ((u+2)/u) * (2*sqrt(x)) du, how can I get rid of 2*sqrt(x)? Replace 2*sqrt(x) by whatever it is in terms of u. 3b: I am still stuck on where the specific # for L comes from, does it have anything to do with the tangent lines? Please see my second October 15 response.

Monday, October 16, 2000 at 09:09:30 (EDT)
MAC 2312-05
Dr M-G, I'm still having trouble with the simpson approximation for the arc length. I have thought of three different ways that might be it but they're all wrong. Could you possibly shed a little more light on this matter? Perhaps the answer immediately below, together with my earlier comments on this question, will help. If not, I'm out of ideas.

Monday, October 16, 2000 at 08:56:11 (EDT)
MAC 2312
i'm confused on 3. (c). the problems asks if both of the inequalities in (a.) and (b.) are satisfied. my confusion is that the inequalities refer to the arc length, but doesn't simspon's rule aproximate the area under f(x)?

i did four-panel simspon's approximation and got 11.772..., that doesn't satisfy the inequalities, but i don't know what to do. can you help?

Sunday, October 15, 2000 at 17:05:36 (EDT)
MAC 2312
First and foremost I would like to express my extreme frustration and dislike for this project.I have a kind of love-hate relationship with mathematics myself. One shouldn't think of hate as being the opposite of love, however, so much as being an intrinsic part of the love-hate cycle. The opposite of love is indifference, and clearly you are not indifferent. So it is possible to view your frustration in a very positive light. Whenever I work on a research problem, there is always a stage when I become extremely frustrated. But the frustration is always worth it, and if I didn't at some stage get frustrated then I would probably never solve the problem. It seems to me that a certain amount of frustration is an inevitable part of problem-solving, all the way through life, regardless of whether you are a baby, a calculus student or a math professor .... And it isn't as though I didn't warn you (remember that picture on my syllabus?—if not, click here). I have been diligently working on it for what seems like forever, and I still haven't found all the answers. Think of the exhilaration you are going to feel when you eventually work it out. And then remind yourself that you couldn't possibly have felt so exhilarated without first of all feeling so frustrated. Just wanted to let you know so you don't think that everyone is having an easy time or that I am leaving things for the last minute....Ok, for Problem #1, when I use cylinders I get 2 pi * the integral(from 0 to 2a)of [ x*sqrt(a^2-((x-a)^2)) dx], and using the washer method,I get pi * the integral (from 0 to a) of [4*sqrt(a^2-y^2) dy]: now, should I directly substitute in x=a+asin theta and y=asin theta like you suggested at this point Yes, because when I did I eventually got sqrt(-sin^2 theta) in the integrals No, you got sqrt(1 - sin^2(q) in the integrals, which is cos(q) and you can't have a negative under the square root Agreedbecause that would involve imaginary numbers, and both integrals were in terms of theta but they still had dx and dy in them.Don't forget that when you change the integral from an x or y integral to aq integral by using a substitution, the integrand changes from whatever it was in terms of x or y to that thing multiplied by dx/dq or dy/dq (which you have to find, of course). Should I be trying to do a u substitution and then directly put in what x and y equal? You should be trying to do aq substitution. I guess that I am asking where to go from here. For Problem #2, a. Is it ok that I proved the second 2 with an appropriate right triangle as well because the "helpful" trig identities were not that helpful to me. Whatever works. c. After factoring the denominator as you suggested and I used the formula you gave on the mock test, do I have that A=u, B=1, sideways fish= (1+sqrt(2)), and beta= (1-sqrt(2))? No (after, of course, replacing x by u in the formula) you have A = 0. Because when I used these, I got an incorrect answer. (I checked it on my calculator) If these values are incorrect, can you help direct me in finding the correct values? For Problem #3, I found the slope of the tan lines at the points using f'(x), but how can I write an Equation for thoses lines when I don't know the y-intercept (and because f(x) is not continuous Although it is scarcely relevant, f is in fact continuous)? Did you look up "tangent line" in the index of your book? See Page 227. Can I graph them on my calculator and guess where the y-intercepts for the lines are? UnnecessaryIs there a more accurate way? Yes And for a., I am still having trouble determining where the specific # for L comes from; does it have anything to do with the trapezoid approximation??? No. Because the graph of f is concave up, if you walk along the tangent line at x = 4 from (4,3) to its point of intersection with the tangent line at x = 9, and from there along the second tangent line to (9,2), you will travel further than if you walk along the curve. Thanks for your time and answers; I realize that I have asked a lot of questions, but I want a good grade on this because I didn't do as well as I thought on our first test.

Sunday, October 15, 2000 at 14:28:54 (EDT)
When we turn in our projects, do you want all the work we have done or just the work that led us to get what we believe to be the correct answer?
The latter.
Friday, October 13, 2000 at 11:35:57 (EDT)
MAC 2312
For Problem #4: b. When finding L, is it better to use the integral formula in the book, or to use the phythagorean theorem example in the book? What, where? Regardless, you have to evaluate an appropriate integral.c. Is the integral of (1/4*x^2) = 1/(8*x)* ln (4x^2)? No. If not, any tips on how to integrate it? Note that 1/(4*x^2) is the same as (1/4) * x^(-2).d. I know that this is simply algebra, but I am stuck on writing y= (1/6)*x^3 + (1/2)x in terms of x. It is in terms of x, so I don't understand the question. Any help in where to start would be greatly appreciated.

Friday, October 13, 2000 at 11:31:42 (EDT)
MAC 2312
For Problem #3: a. The tangent lines, are there two separate lines at each point or one tangent line connecting the two points? There are two separate tangent lines, one for each point. If two separate lines, the point of intersection is where the two lines meet, correct? Correct. If applicable, then, where would the point of intersection be for the one tangent line connecting the two points? N/A Also, as I asked in a previous question, the value for L is pretty specific, do we need to know where it comes from? Yes If so, any hints to where to look? Draw a picture, and note that the curve is concave up.b. Should we show this algebraically or graphically, or does it matter? A bit of both, probably. In any event, use Pythagoras's theorem. Again, for the L, a specific number, same question as in a.Same answer. c. I don't recall ever using a simpson approximation to find L. L is just the definite integral of a particular integrand. A is just the definite integral of a particular integrand, but a different one. If you can do one then you can do the other. So you may be right, but what of it? The assignment is designed to have project-like elements. Is this formula the same for A? No.Actually, just what is the formula?See my answer to your earlier question.

Thursday, October 12, 2000 at 22:12:38 (EDT)
MAC 2312
Some suqestions on how to sub in for 3d? (sqrt(x)+1)/(sqrt(x)-1) ... comming up pretty dry....
Try u = sqrt(x)-1
Thursday, October 12, 2000 at 21:04:50 (EDT)
MAC 2312
when i substitute y=a sin (theta) in problem one of the assignment, how do i get the new limits of integration or do they stay the same?
No, they do not stay the same, at least in general. If the old limits are y1 and y2, then the new limits are theta1 and theta2 such that y1 = a sin (theta1) and y2 = a sin (theta2).
Wednesday, October 11, 2000 at 21:52:15 (EDT)
MAC 2312 Section 6
Hi, my name is (name withheld) and i am in your MAC-2312 ; Section 06 class. I tried to submit you a question on your homework help line but it would not let me submit my question. That's because you attempted to use AOL's server; see above. My question is the following: On our written, take home assignment, question number one about a circle C with radius a that passes through the origin of coordinates and has its center on the positive x-axis. Does this mean that the circles circumfrence passes through the origin? If this is the case, would the volume of the solid we are trying to find be a toriod or something like that or would it still be a sphere? I just can not picture in my mind what the soild would look like, more or less try to find it's volume. Any help at all would be greatly apreciated. Perhaps we could go over this in the beginning of class on Thurs. Thanks for any help, (name withheld)
Here is a picture. Note that both x-axis and y-axis are drawn at a perpendicular distance of 2a from the plane in which they actually lie; furthermore, the y-axis is drawn through x = -2a instead of through x = 0. (The true y-axis is the axis of symmetry.)

Wednesday, October 11, 2000 at 12:53:47 (EDT)
MAC 2312
Question on the substitutions that we are to use to solve both number 1 and parts of 2. These are just straigt subs right? As straight as any I know.Not like ones that we have used to intergrate. Sure, it's a new substitution, but this assignment is designed to have project-like elements. If not I can't understand how sin(x) = 1/sqrt(whatever) can be used because sin just flops between cos and sin. Can you help? For #1, I suggest the appropriate substitution from my October 4 reply below. For #2, just use the substitution given above (a). The relations in (b) are needed to get everything in terms of u, so that you can transform the integral. Once you have done so, you will have to use a partial fraction decomposition to complete the solution. The necessary information is contained in a mock-test formula (or the book—don't forget that you are allowed to look things up in the index) and my October 8 reply below.

Monday, October 09, 2000 at 16:27:02 (EDT)
MAC 2312
On Problem #2: a) For the first 2, do we have to obtain our result by using an appropriate right triangle, or can we do it 2 different ways (including the right triangle way)? For the second 2, what set of notes exactly are these helpful trig identities? I can't seem to find them. b) Can I prove this by graphing, or can you recommend another way? And, (pi/2*sqrt(2)) and (pi/2) seem to be pretty specific numbers, do we need to know where they came from? c)Are we supposed to use the relationships from a) for this?

Sunday, October 08, 2000 at 17:08:15 (EDT)
I've been working on problem #2, letter c from the assignment for the last two hours and I'm on the same point I was two hours ago. I need to integrate (u^2 + 1)/(1-u^2 + 2u) - I have tried factoring the denominator, I have tried finding a substitution, I have tried integration by parts - but I'm sure I did something wrong cause this should be easy, shouldn't it?

Friday, October 06, 2000 at 12:11:09 (EDT)
MAC 2312
For Problem #1: Can I use the equation of the circle as (x-h)^2 + (y-k)^2 = a^2?? If so, to get the circle to go through the origin of coordinates (0,0), the center, (h,k), is at (1,0). I can verify this by graphing on my calculator. When rearranging my original equation for the circle to find what x=, I found that x^2= a^2-y^2+(2*square root(a^2-y^2))+1 because I had h=1 and k=0. Is that correct to do?

Wednesday, October 04, 2000 at 03:09:17 (EDT)
Again on the assignment. Prob. 1 I understand why it can't be done like a cylinder, but I don't want to believe it... look here, this describes the method (http://www.nas.com/~kunkel/torus/torus.htm -- note I found this tonight while I was looking for the equation that describes the volume of a torus), my answer is 4 (pi)^2 a^3... but that's not right is it? Because the line of the circle that is on the origin does not get pushed around 2(pi) like the circle in the torus... but I am lost as where to go.. something to do with the washer method.. but I don't remember the general equation... help if you can.

Sunday, October 01, 2000 at 23:00:02 (EDT)
MAC 2312
Two motorboats on opposite shore of a river start moving toward each other at different speeds. When they pass each other first time, they are 700 yds from one shoreline. They continue to the opposite shore, then turn around and start moving toward each other again. When they pass the 2nd time, they are 300 yds from the other shoreline. How wide is the river? I know this isn't a calculus problem. It's from today's Parade magazine. But I'm in your calculus class, and you didn't say I couldn't ask it. Marylin vos Savant's solution seems like it's from left field. I can follow it, but I would never have though of it myself if I got it on a test. Is there another way? I hate word problems.
Yes, there is another (and, in my view, more natural) way. Hereis how I did the problem (x is the distance between crossing points, v1 and v2 are the speeds of the boats). I suspect it took me less time this way than it took Marilyn (note spelling, incidentally) her way. I had the answer in a minute or so on a piece of scrap paper, where I simplified things by using 100 yards as unit of distance, so that the quadratic equation I actually solved was (3+x)/7 = (14+x)/(6+x). (What did take me considerable time, however, was to write the thing out neatly for you, scan it in, and then compete with thousands of other phone users to upload the file!)
I would argue that my method is simpler and more obvious than Marilyn's. As a bonus, it yields the two different speeds—you were curious about them, weren't you?—without a shred of additional effort. (In fact, it wouldn't surprise me in the least if Marilyn first obtained her answer my way, then rewrote it because even simple algebra is taboo in Parade magazine... But why?)

Thursday, September 28, 2000 at 23:22:02 (EDT)
MAC 2312
Can you do 8.1 # 11. I keep getting 88(something) and the answer is 65(something).
Sure. Click here.
Sunday, September 10, 2000 at 13:22:42 (EDT)
MAC 2312
In section 7.4 #5, I used formula,V.24 for the problem. Why is it that a=the square root of 3 instead of just 3?

Thursday, September 07, 2000 at 20:19:47 (EDT)
MAC 2312
I don't understand how to do number 3, section 7.3 pg 337. Could you please do that one?
Sure. Here it is.
Thursday, September 07, 2000 at 14:48:33 (EDT)
MAC2312 Section 5
Hello Dr M-G! I had a bit of trouble on a few problem and would appreciate it if you can show me the solutions in "DETAIL" to Sec. 7.3 #19 and #16; 7.4 #17 and #19. Thank you!!
Hereis how! in small files with mediocre quality and hereis how in a huge file with much better quality. Note that the answer to #19 of Section 7.3 is identical to the one in the back of the book—it just looks different.
Wednesday, September 06, 2000 at 20:12:19 (EDT)
MAC 2312 Section 5
For Section 7.3 # 23 I do believe I have found the correct parts to integrate the function by but I am having problems with a resulting integral. I have checked the correct answer in the back of the book, what do I need to do to arrive at the same conclusion? Thanks.
Here's how!
Tuesday, September 05, 2000 at 20:41:26 (EDT)
MAC 2312 Section 6
Please do #25 on p. 332.
It's here!
Tuesday, September 05, 2000 at 14:54:59 (EDT)
calculus 2
can you please do numbers 11 and 15 from section 7.2
Sure. Click here
Tuesday, September 05, 2000 at 00:49:15 (EDT)
MAC 2312
On question 1.(b). I could do the problem all the way until I got to the end. I don't understand where the 1/2ln2 came from. I don't understand where they got 1/2 from. I understood the problem up until the final answer. Could you tell me where they got the 1/2 from please. Also could you work #3 out?
Sure. Click here
Monday, September 04, 2000 at 16:54:57 (EDT)
MAC 2312 Section 5
Can you show me how to do number 7 and number 19 on pages 331 and 332 respectively? Thanks.
Sure. Click here
Friday, September 01, 2000 at 15:31:48 (EDT)
MAC 2312 Section 6
Could you please show me how to do No. 45 on p. 327 of the text (Section 7.1)? I can't seem to get the answer in the back of the book.
Sure. Click here
Monday, December 13, 1999 at 19:54:19 (EST)
MAS 3105
Ok get ready ...here we go 1. On the second mock test #2 why again does the vector c=0 make p, q, and r LI (Because three vectors are linearly independent if and only if the only linear combination of them that equals the zero vector is the zero combination) and similar to that on #5 is there a usual formula to see if three matrices, vectore, etc. are LI other than guess work (like A2+A3=2A1)? (A1, A2 and A3 are LI if c1 A1 + c2 A2 + c3 A3 = the 2x2 zero matrix implies that c1, c2 and c3 are all zero, and otherwise LD. So you simply solve the set of 4 linear equations in 3 unknowns that results from c1 A1 + c2 A2 + c3 A3 = the 2x2 zero matrix. If the only solution is c1 = 0, c2 = 0, c3 = 0, i.e., c = transpose of [0,0,0], then the matrices are LI. But if there is a nonzero solution, then the matrices are LD, and the nonzero solution will tell you what the linear dependence is) 2. Same test #4 What would alpha beta and gamma do if the given span did = V (as opposed to alpha+beta+3gamma=0)? (They would be associated with more than just two polynomials, because V has dimension 3, and so you need three polynomials to span it) 3. One more from this test. #6 (i) how did you get from the second to the last step to the last on the solutions (Use the definition with alpha splat x in place of x, etc) and then parts (ii) and (iii)? (If i is the identity then x splodge i must equal x, so x + i + q must equal q, so i must be -q. If y is the inverse of x then x splodge y must equal q, so x + y + q must equal -q, so y must be -2q - x) 4. On the 4th test #2 b) I had given the answer as {[000]}T and you said it was {[oo]} why? [oo] (forgive the notation) (Because the kernel is a subspace of V. So the only things it can possibly contain are 2 x 2 lower triangular matrices) I understand the concept that the dim ker=0 but the dim V=3???(I tell you a basis for V. It contains three vectors. Ergo, dim(V) = 3)

Thursday, December 09, 1999 at 20:51:48 (EST)
MAS 3105
On number 5 of the project, I cannot understand my notes to know how a population matrix is primitive or not. For some reason when I took notes, I did not write them clearly. If you could help that would be great. Thanks :)
I suggest you read the final paragraph of my Notes on Matrix Population Analysis
Thursday, December 09, 1999 at 17:18:02 (EST)
MAS 3105
On the population analysis handout on page four i do not know how you came to the results of 5%, and 50% in the first paragraph. Also on the project assignment, inproblem 5b, what do you mean by the population growth rate, and what calculation do you use to get it?
0.542959/0.517574 = 1.049 implies almost 5% more valuable, and similarly for the next part. The population growth rate is the dominant eigenvalue.
Wednesday, December 08, 1999 at 21:07:29 (EST)
MAS 3105
I do ont understand how you came to the results on the second page of the population analysis handout. What did you do to find the 5.5%, 47.8%, and 46.7% at the bottom of the page?
100 x 0.712253/(0.0815501 + 0.697169 + 0.712253) is approximately 47.8, etc.
Sunday, December 05, 1999 at 02:06:54 (EST)
MAS 3105
How do I calculate the sensativity of the population matrix for the project? I looked at your notes and the equation used is v(i)w(j)=s(ij) but I still do not know how you got your numbers in the sensativity matrix(page5).could you show with a simple example?
From just below (7), w2 = 0. 697169. From (23), v1 = 0.517574. So v1 times w2 = 0.3608 = s12, as required. Etc. Please note the correct spelling of sensitivity.
Thursday, October 28, 1999 at 17:31:32 (EDT)
MAS 3105
Will you please do numbers 1, 15(b), and 15(c) outta sec 4.2.
Saturday, October 23, 1999 at 19:01:58 (EDT)
MAS 3105
Will you please do the one-to-one part of question 11 outta sec 4.1.
Monday, October 11, 1999 at 23:19:51 (EDT)
MAS 3105
According to Theorem 3.7 (pg 172) A nxn matrix is always linear independent. This statement is false. Read the theorem. The columns are LI only if the determinant is nonzero. How come #9 in sec 3.4 is LD? Because the determinant is zero.When asked for LI should I always follow the steps given on page 172?Yes

Monday, October 11, 1999 at 23:13:27 (EDT)
MAS 3105
Could you please do #9,c from sec 3.3.Here it is I tried to compare it to Ex 4(pg164), how does it differ? Basically, it doesn't. But the answer is non-unique. So my matrix B does not agree with the matrix given in the back of the book (however, all possible answers are row equivalent.)

Monday, October 11, 1999 at 23:09:19 (EDT)
MAS 3105
Sorry, that is for sec 3.3. (#11 part a,b,c how would I solve this. For (a) and (b) you need merely observe that the determinants are nonzero (being -6 and 8, respectively). So each set of vectors is a basis for 3-D space, and hence must span. For (c), proceed thusDoes it need to be linear independant?No, because you have more vectors than the space has dimension)

Monday, October 11, 1999 at 23:02:49 (EDT)
MAS 3105
How would I go about to solve #11(a,b,c)? Part c is linear dependent is it not?
Monday, October 11, 1999 at 23:00:03 (EDT)
MAS 3105
Could you please do #10 on sec 3.2
Here it is
Sunday, October 10, 1999 at 22:01:13 (EDT)
MAC 2311
How do you sketch a graph of the derivative function of the given function? Section 2.3 numbers1-9
First identify where the derivative is zero. Plot those points. Then identify intervals on which the derivative is positive or negative (original function increasing or decreasing, respectively), and intervals on which it is increasing or decreasing (original function turning up or turning down, respectively). Use the graph to obtain a rough estimate of the derivative at a few key points (e.g., where it is greatest and least). Then put everything together. Except in the case of No. 1, where the derivative is clearly -2 everywhere, the best you can hope for is a very crude sketch, for example

Saturday, October 02, 1999 at 20:51:18 (EDT)
MAC 2311 Sec10
#19 in sec. 2.2 I can find the derivative, but i can't figure out how to find the equation.
Here's how
Wednesday, September 29, 1999 at 17:53:11 (EDT)
MAS 3105
Will you please do number 6 from section 3.2. Sure

Monday, September 27, 1999 at 19:36:30 (EDT)
MAS 3105
I do not understand problem 17 on sec.2.4. I do not know how to prove this.
Just apply the definition of eigenvalue-eigenvector pair
Monday, September 27, 1999 at 19:33:56 (EDT)
MAS 3105
My question is on Sec.2.4#16. How do I go about proving the statment given?
By straightforward multiplication and addition of matrices
Monday, September 27, 1999 at 19:29:07 (EDT)
MAS 3105
I do not understand what the book is asking for on sec.2.1 #11, this is then related to a problem in Sec.2.4 #12. Could you please tell me what they want me to undetstand from these problems?
If I were you, I'd ignore the bizarre recursive definition of trace and instead define trace as the sum of the diagonal elements. Then what No. 12 of Section 2.4 establishes is that the determinant of a matrix is the product of its eigenvalues and the trace of a matrix is the sum of its eigenvalues. The last part of No. 11 of Section 2.1 is in essence asking you to prove the third part of Theorem 5.1 on p. 304 by an alternative method, for which the previous part of the question is needed. A proof goes as follows
Saturday, September 25, 1999 at 11:42:07 (EDT)
MAS 3105
My question is about Section 2.2, "A Better Way." I do understand Section 2.1 and what I have doing is using those techniques to answer the questions in 2.2, but I would like to know how does the Gaussian Elimination work. When I try I always get the work answer. I also cannot seem to beable to get the Laplace Expansion either. Could you please do two problems for me so I can understand. 3 x 3 matrix (number 1) and a 4 x 4 matrix (number 5). Thank you Here they are

Sunday, September 12, 1999 at 14:19:05 (EDT)
MAS 3105
Sorry to bother you again... I was sick last week and I know you probably answered this last week, but I would really like help on #17 on page 56. (the previous question I asked was on page 54) Thanx
Sorry to hear you've been sick, I hope you are feeling better this week. Here's how to do #17 on page 56
Sunday, September 12, 1999 at 13:37:26 (EDT)
MAS 3105
#2 (b) I know it wasn't homework but if I could better understand what they are asking when they mention multiplying a matrix by I as in 2b I could understand #3 better. Thank you
I think the point of this question is to verify the statements on lines 6 and 7 of p. 60 (three lines below the big matrix)
Wednesday, September 01, 1999 at 17:03:49 (EDT)
MAS 3105
Can you get me started with No. 8 of Exercises 1.4?
How about if I do part (c)?
Wednesday, September 01, 1999 at 17:02:16 (EDT)
MAS 3105
I don't understand the answer to No. 9 of Exercises 1.3 in the back of the book. Can you explain it?
I can but try.
Friday, April 10, 1998 at 20:54:09 (EDT)
MAP 4180, Game Theory
Are you absolutely sure that Kreuzweg is symmetric when tau1 is not equal to tau2?Yes (and it's obvious intuitively, but of course you have to establish it mathematically.) I can't seem to satisfy (1.39).Then you need to think much more carefully about how the reward functions for Kreuzweg relate to f1 and f2 in Section 1.3.

Friday, April 03, 1998 at 18:11:16 (EST)
Game Theory
#4: We define the reasonable set as the vector x such that the ith element of x is less than or equal to the maximum of nu(S) - nu(S-{i}) for all coalitions S. It is important to remember that this statement must hold for all i.. If x is not in the reasonable set, then is the ith element of x strictly greater than max(nu(S)-nu(S-{i}))?For some i, yes.
Tuesday, March 24, 1998 at 16:55:22 (EST)
Game Theory and Application
Dr. M-G, A question about the Nash bargaining solution for problem 3.10. Is it supposed to be a point that is furthest from the origin? Only, in principle, if distance is defined according to p. 102 of Section 3.3; if, in practice, Euclidean distance gives the same answer, then that is merely a coincidence. Can we move this homework back to Friday? Best not to, I think, because the deadline for the following assignment is immovable (but I have no objection in principle, as long as you can live with a tighter deadline for the next assignment). How are we supposed to find this bargaining set? In the book you state that it is (f1-f1tilda)*etc Should we use this formula? But contrary, in the notes, the monday before break, D(u,v|f1 less than or equal to f1 tilda, f2 less than or equal to f2tilda) Could you give an example on how to interpret this?Nothing contrary here. There are two ways to do it. You can either work in the reward set, then find the corresponding point in the decision set (and remember, the mapping's 1-1 in this case); or you can find an explicit expression for dbar in terms of u and v, and maximize it. PS. Scott and Dawn wonder if we can have pizza in class on friday?? Unfortunately, there's a rule against pizza in that room, and I have a family to support.Jayson says a wacka wacka bee achoo. Confucious say move homework back, you have happy student. Confucius also say, tight deadline, big headache.
Tuesday, March 24, 1998 at 13:48:07 (EST)
Game Theory
3.10: I understand that P,R,S,T are arbitrary constants. Do they, however, have the condition of being positive, negative, nonpositive, or nonnegative? Are these assumptions characteristic of the Prisoner's Dilemma?
Good question. On the one hand, the answer is no. On the other hand, suppose S (the least payoff) were negative, say S = -a. Then you could add b = a + epsilon to every payoff to obtain a game with payoffs t = T + b, r = R + b, p = P + b and s = S + b = epsilon. This game would be strategically identical to the original game. So, if I were you, I would assume that all payoffs are positive. It will be easier to draw the reward set, and no generality is lost.
Saturday, March 21, 1998 at 20:20:45 (EST)
game theory
Do you think that you can move the homework back to Wednesday, and leave homework 5 due on the same original date?
Given that Assignment 5's deadline is the one that really concerns me, I guess that's OK. (But I won't move the deadline for Assignment 5. Try me.)
Saturday, March 21, 1998 at 20:10:19 (EST)
Game Theory and Application
A couple of questions.
1. Do we have to use TOA in order to find the solution for 3.10?
No, and I would advise against it.
2. If T>R>P>S how can T&P=0 and R&S=1 as stated in the back of the book?
It doesn't say that in the back of the book. Read more carefully.
3. Can you give us suggestions on as to how to obtain the T,R,P,S?
No, because I don't understand this question. T,R,P,S are arbitrary numbers (except to the extent that they satisfy T>R>P>S).
4. Can you give us some hint as to how to draw the reward set F bar on Mathematica?
No, because in this case using Mathematica would be like using a sledgehammer to crack an egg. And you should never use a sledgehammer to crack an egg.
5. In the hard copy of the book, 3.10a startes that we should find P*, in addition to showing that Pg=P, in the online version of the book, it says show that Pg=P=P*. Which version should we follow.
The hard copy is correct. (Well, what do you expect if you download something for free?)
Monotonous, isnt it??

Saturday, March 21, 1998 at 19:02:38 (EST)
game theory
For problem 3.10, can we assume the information in the back of the book about 1.25, or do you want us to find the rational reaction sets?
Yes, and no. The rational reaction sets are thoroughly irrelevant.

Saturday, March 21, 1998 at 14:02:04 (EST)
game theory
Can we also use 3.15 to check the corner points? If not what should we use?
Yes, of course.
Saturday, March 21, 1998 at 00:50:38 (EST)
Game Theory
Will it suffice to use the Jacobian to find P, or must we find the image of the decision set on the f1-f2 plane and manually narrow it down?No, it isn't necessary to find the image of the decision set. If we can use the Jacobian, how can we eliminate the endpoints? (Refer to Figure 3.2) You can apply (3.15) directly to the points in contention and proceed ad hoc.

Saturday, March 07, 1998 at 19:45:13 (EST)
Game Theory
Could you refresh our memories on the Theory of the Alternative?
Everything you need to know about the TheorEM of the Alternative is on pp. 99-100 of my book, following (3.26). A special case of the theorem suffices: If J is an n x n matrix and K is an m x n matrix, then EITHER there exists a row vector h such that every component of J h^T is positive and every component of K h^T is nonnegative OR there exists a nonnegative row vector mu and a nonnegative, nonzero row vector eta such that eta J + mu K equals the zero row vector, but never both.
Thursday, February 26, 1998 at 20:08:57 (EST)
game theory
After reading your answer to the last question, we realized that we found the wring rational reaction sets, because we thought that u and v went to infinity. Also, you said that the rational reaction set for player one contains the u axis, does that mean one point? Since our solution is obviously wrong do you think you could give us more time so we can ask you questions about it? Thank you
No, actually, the rational reaction set contains more than one point of the u-axis (ask yourself, what is the rational reaction to v = 0?). But I guess you need longer, so Monday's OK, especially given that Assignment 4 now comes after Spring Break.
Thursday, February 26, 1998 at 13:00:39 (EST)
Game Theory
I have a question about problem #2. For the rational reaction set, I found u to maxmize on a line between u and v, and then follow the u-axis for u greater than the u-intercept. But, I was talking with others and they found the max to be the same line only with the v-axis as max when v greater than the v-intercept. In my case I have 3 Nash equilibria, in their's there in only one. Yet, as we analyzed it some more we thought there could be possible infinately many. Could you help us out with this? Would it be possible to tell us how many Nash Equilibria there are?
I don't quite understand what you mean by "maxmize on a line between u and v" or "follow the u-axis for u greater than the u-intercept," but R1 does contain part of the u-axis, and there is only one Nash equilibrium. Nevertheless, there is a sense in which there is almost a second Nash equilibrium. What sense is that? ... You'll have to figure it out. Incidentally, it sounds as though you may be forgetting that (u,v) lies in the unit square and theta is strictly less than 1 (if not, pardon me).
Tuesday, February 24, 1998 at 18:59:09 (EST)
game theory
can you move the homework to friday?
Wednesday, February 18, 1998 at 23:22:09 (EST)
In section 2-8 number 14. I cannot seem to make a diagram for this. I have a parralellagram type drawing but I cannot make a connection to a formula to take a derivitive of..Could you possibly just make a diagram for this problem before the test on Thursday, I know it is last minute......
The diagram is a simple triangle. Measure time from when she starts walking -- from Q, say (where PQ = 500 feet). At that instant, he is 5 x 60 x 4 = 1200 feet north of P. Thereafter, they separate at a speed of 4 + 5 = 9 feet per second in the north-south direction. So suppose the woman is always at W (initially W = P), and that the man is always at M (initially, M = 1200 feet north of P), and draw a right-angled triangle with vertices at M, W and a right angle at, say, R. Then RW = 500, RM = 1200 + 9t and MW = z is the thing whose rate of increase you want to know. Now differentiate z^2 = 500^2 + (1200 + 9t)^2 to get z dz/dt = 9(1200 + 9t) and substitute t = 15 x 60 = 900 and z = 100 Sqrt[8674] to get your answer.
Friday, February 13, 1998 at 21:38:26 (EST)
MAC 3211
Could you please explain how the book arrived at the answer for #7, p.160? To be more specific, part (a) 15/6 = x+y/y, y = 2/3x ???
So far, so good. The man's speed is dx/dt = 5. So y = 2x/3 implies dy/dt = 2/3 (dx/dt) = 10/3. The shadow is distance x + y from the pole. So (a) the speed of its tip is d{x+y}/dt = dx/dt + dy/dt = 5 + 10/3 = 25/3 and (b) the length of the shadow is y, so it is lengthening at rate dy/dt = 10/3.
Sunday, February 08, 1998 at 10:17:21 (EST)
Game Theory
Im confuseed on SWII. How in the world did you find Eqns 1.78(a-c)? Im getting some really kooky fractions for the probabilities, are they supposed to be like this, or are they nice fractions?
You get the first, second and third of (1.78) by maximizing the first, second and third of (1.74) with respect to u, v and z, respectively. I don't know about your kooky fractions, but (1.78) is correct.
Ok, for SW, 1.11, when we redefine alpha, does that mean that we redefine all of the boundary lines such as u + 1, u + 6 etc? or we are just shrinking the box and the same boundary lines apply?
No to the first. Yes, reducing alpha merely shrinks the box from the size you see in Figure 1.11 to the size you see in Figure 1.12. Nevertheless, it completely changes the implications of calculations you did for Region C in Exercise 1.9. You will need to reinterpret those results (drastically); however, you should not need to do the calculations all over again.
Saturday, February 07, 1998 at 17:57:30 (EST)
Game Theory
Do you think you can move back the homework deadline to Wednesday?
Yes, but no later than Wednesday.
Saturday, February 07, 1998 at 17:56:39 (EST)
Game Theory
For some reason the rest of my question didn't get through, but as I was saying I found out the max to be u+1 for u between 9/2 and 9, but now I don't know how to figure out whether the max is u+6 or u+1 for all other u.
In fact, it need be neither. Click here for more details. Incidentally, it is best to avoid inequality signs when you ask a question, because they are special symbols in HTML.
Thursday, February 05, 1998 at 00:50:31 (EST)
Game Theory
Well I think that no one asked a question in class because we were all confused when you just walked in and just started teaching. Usually, you will ask us if we have any questions, and or gie us a little "dead air time" to start asking, I know that I, if not other people were just confused as to what the hell you were doing today. Could you please go over somehting from 1.6, like how you got the reward functions for the 3 player system?? Your friend.. A game theory student, concerned about the welfare of his/her fellow classmates
Please see Note 1. The second paragraph of my Teaching Statement is also relevant.
Wednesday, February 04, 1998 at 14:17:57 (EST)
Game Theory
Is it really fair to have us do a problem on section 1.6 when it wasn't covered in class and from what I heard today, nobody understands it?
Yes, of course, it is fair. Moreover, it's my duty to challenge you. If I covered Section 1.6 in class, then I would - in effect - have done Exercise 1.21 for you. Confucius say (something like) a negative attitude can make the simplest of tasks seem impossible; correspondingly, a positive attitude can make the most daunting of challenges seem trivial. Don't forget that anyone can ask a question at any time (and everyone knows what Confucius says about questions). If "nobody" understands, why has no one asked a question about Section 1.6 in class?
Monday, February 02, 1998 at 01:26:50 (EST)
Game Theory
This homework assignment is lengthy, could we have another week to work on it?
You would probably consider all of my assignments lengthy (but they're good for you). OK, I'll extend the deadline to February 9, that gives you another week.
Tuesday, January 20, 1998 at 18:40:27 (EST)
Game Theory
Will the homework be due on Friday, or will you move it up to Monday?
No to the first, yes to the second.
Monday, January 19, 1998 at 23:30:59 (EST)
Game Theory
Could you please tell me how to find v1 and v2 in table 1.8 and the one that we did in class? Thanks.
Huh? ... What kind of a question is that? Please see me during office hours.
Monday, December 01, 1997 at 15:13:47 (EST)
Is the mock test for the final both sets of questions on your web site?
The final could be like either set of questions, or any combination thereof.
Monday, November 17, 1997 at 14:43:49 (EST)
Is there any curve on the final class grades?
The best answer is no. Please talk to me in person if you require further elaboration.
Tuesday, November 04, 1997 at 12:27:38 (EST)
I got Assignment C off the internet and it is a little hard to read.
You should come to class and get a hard copy.
In #3 is it (Bt-C) / t^s?
No, it's (Bt-C)/ t^5.
In #4 is the fraction 1/65?
No, it's 1/63.
Monday, November 03, 1997 at 20:16:56 (EST)
What defines a p.d.f. and how does one find a p.d.f. given a c.d.f.?
See old Lecture 14 (new Lecture 19).
Tuesday, October 14, 1997 at 17:07:42 (EDT)
MAC 3311 M, 9:05-9:55 T,R 9:30-10:45
I have a question about number 1, part 3 on the second mock test. We are supposed to find the values of a and b. I did the part where
W(2-) = W(2+)
W'(2-) = W'(2+)
and I got
8a = 4/(b-2)
12a = 4(b-1)/(b-2)^2
From there I looked at the answers posted and saw that you divided. I don't understand why. I also tried to solve for c and b in terms of a in Exercise 10.3 in the lecture notes and I couldn't solve it. Could you tell me how you would solve for the variables in these circumstances and why?
There are two ways to eliminate a from these equations. One is to divide them, so that a cancels on the left-hand side and 4/(b - 2) cancels on the right-hand side, leaving 2/3 = (b - 2)/(b - 1) = (b - 1 - 1)/(b - 1) = 1 - 1/(b - 1), which implies 1/(b - 1) = 1 - 2/3 = 1/3. Hence b - 1 = 3, or b = 4 (implying 8a = 4/(4 - 2) = 2, or a = 1/4). The second way is to multiply the first equation by 3/4 to yield 6a = 3/(b-2) and to divide the second equation by 2 to yield 6a = 2(b-1)/{(b-2)^2}. Then (because obviously b is not equal to 2 or else a could not be finite) 3/(b-2) = 2(b-1)/{(b-2)^2}, implying 3(b - 2) = 2(b - 1) or 3b - 6 = 2b - 2, which simplifies to b = 4. I used the first method because I thought you would be more likely to think of the second method, and then my solutions would be more informative. But either method will do. Whichever you think of first is best. As for equations (38) in old Lecture 10, adding (38a) to (38b) yields 2A + 4B + A + 4B = 1 - C/4 + C/4 = 1 or 3A + 8B = 1, so that B = (1 - 3A)/8. Subtracting (38b) from (38a) yields 2A + 4B - (A + 4B) = 1 - C/4 - C/4 = 1 - C/2 or A = 1 - C/2, so that C = 2(1 - A) on rearranging. In this case the best answer to why is probably that it is just about the only way; division doesn't help in the least!
Tuesday, October 14, 1997 at 11:13:01 (EDT)
I can neve seem to find the answers for Assignments. They are not under "Assignments, Tests, and Solutions" where am I supposed to look for them?
I beg to differ. The answers most certainly are under "Assignments, Tests, and Solutions" ... all you have to do is click on Fall 97.
Saturday, October 11, 1997 at 23:44:46 (EDT)
Will table 16.2 be a given on test 2, or should I memorize it, or can we not use it?
In general, you should never memorize anything. If it is worth remembering and you have used it enough, then you will remember it automatically. Therefore, in particular, (i) if you cannot remember something that is worth remembering, then you haven't done enough homework problems and (ii) you shouldn't memorize Table 16.2. (In any event, you won't need it on the second test, and so it will not be given.)
Wednesday, October 08, 1997 at 22:24:32 (EDT)
I am a bit confused on the error term for Assignment B number 2. What am I supposed to be finding and how do I go about it?
Please see my reply to the earlier question (below).
Wednesday, October 08, 1997 at 19:06:22 (EDT)
I am still having problems with 3 and 4 on Assignment B. The answer for 10.13 I still cannot find under mac3311.s97.html. I seem to be getting stuck on every aspect that I try. Can you help me?
There's a solution to old Exercise 9.13 in the solutions manual at the end of the course pack. Note the typo in the first line of the solution (t squared, not t cubed, in the denominator). There are several relevant solutions here For example, the second solution on Assignment B for old Exercise 9.16; the fifth solution on Assignment B for old Exercise 10.13; the first solution on the second test for old Exercise 10.16; and the final for old Exercise 10.12. Together, these examples should more than suffice for Problems 2-4.... And isn't this a bit late to be asking for help?
Thursday, October 02, 1997 at 01:57:54 (EDT)
Can you give me a hint for problem #5 on Assignment B? Maybe you can make up one exactly like it so I can go through it step by step to understand.
Words are all I have.
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