ANNOUNCEMENTS
MTH 254 — Fall 2010

12/12/10
I expect to be in my office tomorrow (Monday) from roughly 1–3 PM.
12/9/10
Grading done; grades uploaded. The average on the final was 68/120.
Clearly the exam was harder than I had intended. Hopefully, the grading scale compensates for this.
Exams can be collected from me; if I'm not in my office tomorrow, you can try next week or wait until next term.
12/8/10
Done grading, but still adding and recording. I expect course grades will be determined tomorrow, and should be available Friday via OSU Online Services.
I should be in my office tomorrow afternoon (Thursday) starting around 2:30 PM.
12/7/10
Still grading; should be finished sometime tomorrow; stay tuned...
12/6/10
Below are the answers to the final. Full solutions can be seen in my office later this week (or next term).
1. 15π/4
2. 0
3. (a) vector; connect arrows (b) scalar; project
4. (a) 4/5 i + 3/5 j (b) straight line
5. ax+by−(2a+3b)z/6=d (for any values of a,b,d; many solutions)
(NOT: 2x+3y+6z=6, which was the answer to the quite different question on the midterm.)
6. x+4y=9
7. −2x i + j
8. (a) (2i+j+2k)/3 (b) 3 deg/ft (c) 0 deg/ft
9. r
10. max is 16; min is −2
11. (−1,1,1), (1,−1,1), (1,1,−1), (−1,−1,−1)
12/2/10
Extra office hours this week:
Don: 1–3 PM on Thursday, 12/2
Carrie: 1–2 PM on Friday, 12/3
If you borrowed a copy of the textbook from the publisher at the beginning of term, those books need to be returned in the MLC on Monday, 12/6, from 12–5 PM, or on Tuesday, 12/7, from 9 AM–12 PM.
Students taking MTH 255 next term may keep the book to use in that course.
12/1/10
The final will be Monday 12/6/10 from 7:30–9:20 AM.
Section 010 (9 AM) will take the final in Weniger 153.
Section 020 (10 AM) will take the final in Peavy 130.
11/30/10
Problem §12.8:51 is challenging. Here are some hints:
Method I: Use the result of problem §12.1:86b to obtain the function to minimize.
Method II: Treat the point on the surface as constant, and write the (squared) distance as a function of the x and y coordinates of the point on the plane. Minimize this function of two variables, thus deriving the formula given in §12.1:86b by another method. Now let the point on the surface vary, and again minimize the (squared) distance (as a function of two variables).
Method III: Treat the (squared) distance as a function of four variables, namely the x and y coordinates of both the point on the surface and the point on the plane. Set all four partial derivatives to 0 and solve.
(The algebra is messy, even with the help of a computer algebra system...)
11/29/10
Here are the figures shown in class today:
Example 1 shows the graph of x2+xy+y2=3, together with level curves of x2+y2.
Example 2 shows the graph of x2+y2=9, together with level curves of xy.
You should recognize the second example as the boundary problem discussed last Wednesday, which leads to two more ways to solve the boundary max/min problem:
Boundary, Method III:
The function being maximized is h=xy, and the constraint is g=x2+y2=9. The gradients of these two functions must be parallel at a local max/min on the boundary, so we compute ∇h=yi+xj and ∇g=2xi+2yj. Setting ∇h=λ∇g implies that y=2λx and x=2λy, which leads to y=4λ2y, so that λ=±½ and y=±x.
Boundary, Method IV:
As above, but set ∇h×∇g=0. Calculating the gradients as before and computing the cross product results in (2y2−2x2)k, which can only be 0 if y=±x.
In either case, you should recognize the four critical points as the points where the level curves of xy are tangent to the circle of radius 3, as shown in Example 2.
11/24/10
Several students have asked me to post the example from class today:
Find the max/min values of the function h=xy on the disk x2+y2≤9.
Critical points:
Differentiate, obtaining ∇h=yi+xj. Setting this equal to zero, the only critical point is (0,0).
Boundary, Method I:
The boundary of the disk is the circle given by x2+y2=9. Solve this equation for y, obtaining y= ±√ 9-x2   , so that h= ±x√ 9-x2   . Taking the derivative leads to dh/dx=±(9-2x2)/ √ 9-x2   , so there are critical points on the boundary at x=±3/√2 and x=±3, corresponding to the points (±3,0), (±3/√2, ±3/√2), (±3/√2, ∓3/√2).
Boundary, Method II:
The circle of radius 3 can be parameterized by x=3cosφ, y=3sinφ, so that h=9sinφcosφ=9sin(2φ)/2. Taking the derivative leads to dh/dφ=9(cos2φ-sin2φ)=9cos(2φ), so there are critical points on the boundary at φ=π/4,3π/4,5π/4,7π/4, corresponding to the points (±3/√2, ±3/√2), (±3/√2, ∓3/√2).
Table of Values:
Using either method, one now evaluates h at each of the points found above (including (0,0)), and discovers that the max/min values of h are ±9/2, respectively.
11/22/10
There is a (not very enlightening...) derivation of the Second Derivative Test at the end of Appendix B in the online version of the text.
A simpler derivation can be found in some other textbooks; contact me or stop by my office if you'd like to take a look.
An alternative derivation, which however requires some basic knowledge of eigenvalues and eigenvectors, can be found in our book.
11/19/10
Don Hickethier's office hours (including MLC hours) are canceled from now through Monday 11/29, inclusive.
I am available for appointments on Tuesday afternoon, 11/23. Other times may also be possible, including that Tuesday morning (before 10:30 AM) and late Monday or Wednesday afternoons.
11/18/10
UPDATE: You can download a copy of today's activity here.
One way to solve this week's homework problems can be found here
Let me know if you left something in Weniger 212 this morning; I may have it.
11/17/10
There is a typo in the homework (now fixed): "R2" should have been "R2"; sorry about that...
Here are the figures shown in class today. In each case, the first figure shows the (3-d!) graph of a function z = f(x,y), and the second shows the combined (2-d!) graph of the level curves and gradient of f.
paraboloid: graph gradient
saddle: graph gradient
11/16/10
Congratulations to Don Hickethier, who successfully defended his doctoral dissertation this afternoon, and will receive his Ph.D. as soon as the paperwork is completed.
11/15/10
Don Hickethier's office hours tomorrow (Tu 11/16) have been changed to 11 AM–1 PM.
Wish Don luck — he defends his doctoral dissertation that afternoon. Further details are available here.
11/14/10
The second quiz will be in class on Friday, 11/19. It covers the material from the midterm through Wednesday's class (11/17/10), namely partial derivatives up through chain rule, as well as the basic properties of the gradient and directional derivatives. This material corresponds roughly to §12.4–12.6 in the text.
The grading policy clearly states that a 3″×5″ index card of notes is permitted on all exams, including quizes.
Yes, you may bring your two previous cards as well, but you shouldn't need them.
11/12/10
All recitations next Thursday (11/18) will meet in Weniger 212.
11/11/10
In addition to my regular office hour tomorrow morning (11:15–11:45 AM), I will also be available in the afternoon for those wishing to discuss their grade prior to the deadline to withdraw (11:55 PM Friday 11/12).
I expect to be in my office from 2:30–3:30 PM Friday; later times may also be possible if you contact me in advance.
11/8/10
Below are the answers to the midterm. Full solutions can be seen in my office.
1. (a) 4 (b) 0 (c) 0 (d) 2k
2. 2x+3y+6z=6
3. (a) positive (b) need more information
4. (a) ∫03 ∫02π ∫25 r dr dφ dz (b) ∫0π/4 ∫0π/2 ∫05 r2sinθ dr dθ dφ
5. 45
6. 61
7. x=3
10/29/10
The midterm will be Friday 11/5/10 in class.
10/28/10
You can download a copy of today's activity here.
10/27/10
A JAVA applet which illustrates the geometry of the cross product can be found here.
10/22/10
All recitations next Thursday (10/28) will meet in Weniger 212.
10/21/10
An image showing how to embed a tetrahedron in a cube can be found here.
10/20/10
A JAVA applet which illustrates the geometry of the dot product can be found here.
10/18/10
If you are having difficulty evaluating some of the single integrals which arise when doing recommended problems from the text, feel free to look them up in integral tables, such as those in the back of the book.
It is not my intention to challenge you on exams with difficult single integrals.
10/17/10
The first quiz will be in recitation on Thursday, 10/21. It covers the material through last Friday's class (10/15/10), namely multiple integration. This material corresponds roughly to §13 in the text.
The grading policy clearly states that a 3″×5″ index card of notes is permitted on all exams. Yes, this includes quizes.
10/15/10
The hemisphere example we did in class today had constant density; this is not always the case. When computing center of mass, don't forget that both integrands should contain the mass density.
10/14/10
You can find out more about the reasons for the choices discussed in class for the names of the spherical coordinates in our paper:
Spherical Coordinates, Tevian Dray and Corinne A. Manogue, College Math. J. 34, 168–169 (2003)
a copy of which is posted on my bulletin board. The short answer is that most students will need to switch conventions at some point during their education, so this might as well be done sooner rather than later.
10/8/10
A copy of today's worksheet on volume elements in curvilinear coordinates can be found here.
10/7/10
Discussions with several students suggest that many of you are attempting to do the assigned homework problems without first trying simpler problems, either through the online resources provided by the publisher (MyMathLab), or directly from the book. As discussed both the first day of class and in an earlier announcement, this strategy is unlikely to be successful for most students.
The assigned homework problems are not intended to be straightforward.
Mastery of calculus requires lots of practice. On the other hand, I do not believe in either assigning or grading numerous drill problems. It is your responsibility to attempt a sufficient number of such problems to develop the necessary skills. Of course we will help, and we welcome questions about specific problems — that's part of what our office hours (and the MLC) are for. Alternatively, immediate feedback is available if you use the online resources.
10/2/10
My afternoon office hours are canceled both this week and next.
My morning office hours will still be held (covered by Don Hickethier).
Check with your TA (and/or with Don) to arrange an appointment if the remaining office hours do not work for you.
9/29/10
A set of keys was found after our lecture this morning. They have been left with the receptionist in the Math Dept office.
9/28/10
Here are the promised Course IDs for MyMathLab:
math97585 (This course is labeled MTH 254, and contains most of the recommended problems from the study guide.)
math05057 (This course is labeled MTH 251, but contains all the available problems from the entire book.)
We are not covering the material in the same order as the book; the MTH 254 "homework" assignments will not be in the right order.
9/27/10
A list of differentiation rules in differential notation can be found here.
You can integrate both sides of each of these rules. The first block of rules then gives you the most important basic integration formulas, while the next-to-last rule gives you integration by parts.
Don't forget that integrating "du" gives you "u" back (up to constant)!
9/26/10
Here is some information about MyMathLab, the online resource center provided by the publisher of our textbook.
9/24/10
Office hours for the TAs have been aded to the course homepage.
At the TA's discretion, you may make use of the office hours posted for TAs other than your own.
Please identify yourself as not being in their class.
9/9/10
My office hours are posted on the course homepage. Clicking on the calendar icon on that page will bring up my full weekly schedule, which is also available here.
9/5/10
Make sure you read the note about textbooks, and take a look at the grading policy.