Math 250B (Spring 2010) HW Assignments and WebAssign Updates

• Final Grades - (posted 5/15) Are now posted an should be accessible via UAccess. You can also see your final scores via D2L. Overall, everyone did really well and the effort going into the final paid off for most. A couple people were right on the edge and the extra credit HW assignment at the end (graded out of 25 points, and contributing up to an additional 3% towards your final grade) helped push some over the edge and up a grade. Unfortunately, there were several others who would have benefitted a letter grade if they had done that assignment, but did not. Keep that in mind down the road. Let me know if you have questions. Am happy to discuss details of your grade/scoring if requested (I want things to be completely transparent), but will discuss particulars in person only so to make sure your privacy is preserved. Via email, one never really knows who they are communicating with.
  
  Let me know if you have questions, but again, I want to say congratulations to everyone. You worked hard and I think that work is ultimately going to pay off down the road. Also, if you want to just for the hell of it, go ahead and rank me via the following website . Anonymous, and would be helpful to me to see what you guys have to say in a more 'casual' forum.
  
  Good luck to all down the road!! Has been a pleasure guys......
  
  
• Final Exam Results - (posted 5/14) The final has been graded and the scores posted on D2L. Overall, most did really well! Excluding the extra credit, the mean was 84+/-9%. But with the extra credit, the mean was 91+/-11%. 17 people got a score on the final over 90% with 6 above 100%! Great job! High score was 110.5% and the low was 65%. Also, for those (excused) who missed a mid-term exams, your 'replacement' score stemming from the final has been posted on D2L. Still waiting some grade files from Joseph, but hopefully I'll be able to get the final course grades posted something within the next few days. Again, overall, great job everyone!
  
  
• Final Exam (05.14.10) - (posted 5/8) The final will take place the morning of Friday 5/14 from 8-10 AM in our usual room (ILC 125). It will cover material from the whole of 250B (save for fundamentals from 250A that carried over directly). Below includes an outline of the salient topics (basically culled from the midterm reviews, plus several new topics since the last in-class exam) covered during the course and thus fair game for the final. As it stands now, there are 6 problems on the final plus two extra credit problems (though that is subject to change). You should hopefully have ample time to work on the exam within the two hour timeframe: if you really know your stuff, it should take in the ballpark of an hour and some change. One suggested strategy for studying is to spend some time looking over L.L. ch.10 (on nonlinear systems). The approaches to handling nonlinear systems really builds off of a lot of the core concepts from the course, so having a good grasp on nonlinear systems will really help cement things together. But keep in mind that the final is comprehensive, thus there potentially will be problems on a wide variety of topics such as: improper integrals, infinite series, power series and their convergence (including Taylor and Fourier series), linear independence, linear systems (e.g., finding eigenvalues and the general solution), .....
  
  Three other items of note. First, while not required to complete the final, you will be able to use the classroom PCs on the exam to access pplane (and pplane only! web browsing otherwise is strictly not allowed). So keep that in mind while preparing. Second, there will be a review session run by Joseph on Wednesday evening 5/12 from 6-8 PM in Math 102 (not the usual 101!). This session should provide you an opportunity to ask any last minute questions as well as work on some various review problems. Third and last, I will not be holding my usual office hours during finals week. However, I am happy to try to schedule an appointment with you if you want to come in and talk/ask questions. So feel free to send me an email if needed.
      
       - Solving ODEs via separation of variables   H.H. 11.4; L.L. 4.1
       - Evaluating Improper Integrals   HH 7.7
       - Convergence of Improper Integrals (e.g., comparison test)   HH 7.8
       - Sequences, Infinite Series   HH 9.1, 9.2
       - Geometric Series   HH 9.2
       - Convergence of Infinite Series   HH 9.3, 9.4
      NOTE: There are lots of good 'practice' problems at the end of Chs.7 and 9 (in the review section).
      
       - Power Series   HH 9.5
       - Intervals/Radii of Convergence for Infinite Series (incl. Taylor series)   HH 9.4, 9.5
       - (Approximating functions via) Taylor Polynomials   HH 10.1
       - Taylor Series   HH 10.2, 10.3
       - Using Power Series to solve Integrals and ODEs   LL 1.4, appendix A.3; class notes
       - Fourier Series   HH 10.5; class notes (just be comfortable with the basic idea, though you won't need to do any calculations)
       - Binomial Series, Binomial Coefficients   HH 10.2; class notes (also see this link for a general overview)
       - Linear Independence, the Wronskian   LL 8.4; class notes
      
       - Classifying types of ODEs (including higher-order equations) [e.g., homogeneous vs. non-homogeneous]
       - Harmonic oscillator   LL 6.4
       - Complex Numbers   HH Appendix B; LL Appendix A.4
       - 2nd Order Homogenous ODEs (e.g., eigenvalues, characteristic equation)   LL 6.2, 6.3
       - Existence & Uniqueness Theorem   LL 6.2
       - Linear, Autonomous Systems of 1st Order ODEs (including matrix form)   LL 6.2, 9.1
       - Phase Plane Analysis   LL 6.5, 9.2
       - Converting 2nd Order ODES to 1st Order Systems and vice versa   LL 6.2
       - Applications in chemistry and population dynamics   see 4/6-7/10 class notes
      
       - Non-Homogeneous 2nd Order ODEs   LL 7.1
       - Method of Undetermined Coefficients   LL 7.2
       - Harmonic oscillator w/ sinusoidal forcing & Resonance   LL 7.3 (especially ex.7.17)
       - Cauchy-Euler Equation   LL 8.5
       - Nonlinear systems   LL 10.1
       - Linearization of nonlinear systems   LL 10.3
       - Nonlinear modeling applications   LL 10.4
  
  &rArr Problems at the end of the sections listed above provide an excellent means for preparing for the final!
  
  
• Quiz 10 solutions here
  
  
• HW XI due 5.1.10 (at 5:59 PM) - (posted 4/23) This FINAL assignment is a combination of WebAssign problems as well as some problems from the book. As for the latter, the book problems are optional and will be counted as 'extra credit' (though they are pretty cool, so you are encouraged to do them!).
  
  If you decide to do the book problems, you will need to hand in a hard copy of these by the last day of class, 5/5/10. Problems from the book (Lomen & Lovelock) are as follows:
  2.3: #s 2, 14
  6.7: #s 3, 5a
  7.3: #s 1, 2, 6, 12 ('The Running Lizard'!)
  
  The more of the problems you tackle (i.e. try to tackle them all!), the more additional extra credit you can receive (which can help towards your final grade). Doing them will also help get ready for the final exam.
  
  NOTE that there will still be a quiz in class on Friday 4/30/10 (the last of the semester) based upon the WebAssign problems.
  
  
• Lab II The second (and final!) project has been posted (pdf) and will be due Monday, 5/3/10 at 5:00. This project focuses on the deriving the Michaelis-Menten equation via some dynamical systems theory. You will need to deal with a system of nonlinear ODEs by way of the Law of Mass Action. The goal is to provide a gentle introduction as to how systems of ODEs can be applied towards understanding a very fundamental problem in the physical sciences.
  
  
• UPDATE (4/16/10): Exam 3 has been graded. I know this was a tough one, but most of you persevered and are all the better for it (whether you believe me or not). Given that it was a tough/long exam, the average scores were fairly low, so I made a uniform upward curve and added 12 points to everyone's score (thus calibrating the top score to 100%). This shifted the average to ~80% (with a median of ~83%). The distribution was well peaked in the range of 80-90%, but there was a long tail and thus the standard deviation (13%) is likely artifactually high. In terms of distribution breakdowns, the #s are as follows: <60(E): 2, 60-70(D): 4, 70-80(C): 7, 80-90(B): 10, 90-100(A): 5. Overall, I think most people did decent. For those who had a tough time and have a score on the low end, it may likely be a good idea to come see me so to come up with a battleplan going into the final. You guys have been working hard and I believe it is going to ultimately pay off down the road.
  
• HW X due 4.24.10 (at 5:59 PM) - (posted 4/16) The tenth HW assignment is now up. It deals primarily with non-homogeneous second order ODEs. L.L. chapter 7 should serve as a useful reference (e.g., the Method of Undetermined Coefficients).
  
• Quiz 9 solutions here
  
• Exam III (04.16.10) - (posted 4/9) The third exam will be next Friday morning (4/16). Any material we covered since the last exam (i.e., after 3/8) is fair game for the exam. The best bet is to look over the course notes carefully. The list below lists salient topics as highlights (and includes relevant book chapters from Lomen & Lovelock).
       - Classifying types of ODEs (including higher-order equations) [e.g., homogeneous vs. non-homogeneous]
       - Harmonic oscillator   LL 6.4
       - Complex Numbers   HH Appendix B; LL Appendix A.4
       - 2nd Order Homogenous ODEs (e.g., eigenvalues, characteristic equation)   LL 6.2, 6.3
       - Existence & Uniqueness Theorem   LL 6.2
       - Linear, Autonomous Systems of 1st Order ODEs (including matrix form)   LL 6.2, 9.1
       - Phase Plane Analysis   LL 6.5, 9.2
       - Converting 2nd Order ODES to 1st Order Systems and vice versa   LL 6.2
       - Applications in chemistry and population dynamics   see 4/6-7/10 class notes
  
  &rArr Problems at the end of the sections listed above provide an excellent means for preparing for the exam!
  
  
• Quiz 8 solutions here
  
  
• HW IX due 4.12.10 (at the end of class) - (posted 4/2) This ninth HW assignment unfortunately has little coded in via WebAssign. Thus, in addition to the two problems posted on WebAssign, you will primarily be asked to do a set of problems from your ODE text (Lomen & Lovelock). These will need to be written up by hand and turned in on Monday, 4/12/10 at the end of class. No late HW assignments will be accepted! A random subset of these problems will be graded and comprise 1/2 of this week's HW grade. As usual, there will be a quiz on Friday 4/9 that is taken directly from the assigned problems. The problems you are expected to hand in are:
    - 6.2: 2(a-h), 6, 13*, 14(a-c), 19
    - 6.3: 1(i,j,k,l,m,o), 6, 7, 11
    - 6.4: 7, 9, 12
    - 6.5: 3, 6, 8
  * - seems like I goofed up a bit in lecture on 4/2....
  
  
• Lab I (Fourier) solutions here (please don't distribute!)
  
• Quiz 7 solutions here
  
• HW VIII due 4.3.10 (at 5:59 PM) - (posted 3/27) The eighth HW assignment is now up. It deals primarily with homogeneous second order ODEs, though a few non-homogenous ones snuck their way in. There are also a few problems dealing specifically with the 'harmonic oscillator'. NOTE: you should be able to input oscillatory solutions as either sinusoids or complex exponentials (though I did not test this exhaustively!) using i as the symbol for the square root of -1.
  
• Lab II - Discussion Assignment 2 - (posted 3/4/10) For the the second lab, you will actually be visiting the Applied Mathematics Laboratory (AML) and Biophysics Lab (BPL) in the basement of the physics building (PAS; on the north side of the building). You are expected to attend one of two sessions on a given date: Tuesday 3/23 3:15-4:15 OR Wednesday 3/24 2:00-3:00. Students have been assigned into groups for a given day, which are indicated below. If you have not yet been assigned into one of the two groups, contact Joseph or I immediately.
  
  + The easiest thing to do will be to meet in front of the physics building (PAS) at the central north-side entrance (on 4'th street) five minutes before the scheduled time. For Group 1, we can simply walk over together after class on 3/23. If you somehow miss the group, enter PAS on the north side, taking the stairs down, making a left on that basement floor, the labs are a few doors down on the left.
  
  + Similar to the first reading assignment, you will be expected to participate in an online discussion forum after the lab visit. Your grade for the assignment will contribute towards 'Reading & Discussion Assignments' portion of your final grade (as described on the course website). You will be graded as follows: you are expected to add a minimum of two substantial posts to the forum. Each post will be accessed on a scale of 1 to 3: 1-little content/contribution to the post, 2-adds in a point to the flow of the discussion, 3-makes a significant addition or insight to the discussion. For most, this grading scheme will be somewhat moot: people tend to get engaged and a fruitful discussion flows fairly naturally (thus such participants will readily receive full credit).
  
  + The basis for the online discussion will be similar to the reading assignment: How does the mathematics we have be exploring in class relate to the experiments you observed in the lab? For example, what mathematics are associated with Brownian motion? Or what types of mathematical tools would one use to describe the fluid dynamics experiments?
  
  + You are encouraged to pose/answer questions and do some additional research into the various phenomona you observed in the lab session (e.g., look up 'BZ reaction' in wikipedia). The course website also contains some additional information (e.g., pictures from the lab visits) to help stimulate some thoughts/ideas.
  
  + A handful of pictures to help remind you of some of the experiments:
       o Lab overview
       o Computer vision
       o Fingers
       o Liquid crystals
       o Laser tweezers
       o Branching
       o Brownian motion
  
  
  + You should be able to access the forum via logging onto D2L (it should be officially open as of 6 PM on 3/23/10). If you have trouble, contact me (CB). The forum will close for additional posts on 4/9/10 at 5:59 PM, so consider that the official DUE DATE.
  
  + UPDATE (4/14/10): Summary of the student posts can be downloaded here
  
  
• HW VII due 3.27.10 (at 5:59 PM) - (posted 3/12) The seventh HW assignment is now up. It deals with finding solutions to second order ODEs (see LL 6.2 and 6.3 for reference) and complex #s (see HH appendix B for reference). We will have a quiz on this material on 3/26. Note that the first lab (Lab 1 - Fourier Series) is also due on 3/26!
  
• Quiz 6 solutions here
  
• Lab I The first project has been posted (pdf) and will be due Friday, 3/26/10 at 5:00. This project focuses on the Fourier series and an application of such. Note that you will be required to use a computer to record a sound file (of your own voice) and subsequently use Matlab to do some analysis upon such. Make sure to give yourself plenty of time here in case you hit any technical bumps in doing this portion of the lab (feel free to come talk to either Joseph or I, but do not wait until the last minute!!).
  
• No HW This Week - (posted 3/1) Due to the upcoming exam on 3/8 (see below), no HW assignment this week (and thus no quiz come Friday 3/5).
  
• Exam II (03.08.10) - (posted 3/1) Our second exam will be next Monday morning (3/8). We have covered a lot of ground since the last exam. Use below as a guide only (also including relevant book chapters to look at for reference, HH for the calculus book and LL for the ODE book). The problems at the end of Ch.9 and 10 in your HH book are also good for practicing basics of series convergence and Taylor series. You will be expected to know the Taylor series expansions for the following functions: sin(x), cos(x), e^x, 1/(1-x), and ln(1+x).
       - Power Series   HH 9.5
       - Intervals/Radii of Convergence for Infinite Series (incl. Taylor series)   HH 9.4, 9.5
       - (Approximating functions via) Taylor Polynomials   HH 10.1
       - Taylor Series   HH 10.2, 10.3
       - Using Power Series to solve Integrals and ODEs   LL 1.4, appendix A.3; class notes
       - Fourier Series   HH 10.5; class notes (just be comfortable with the basic idea, though you won't need to do any calculations)
       - Binomial Series, Binomial Coefficients   HH 10.2; class notes (also see this link for a general overview)
       - Linear Independence, the Wronskian   LL 8.4; class notes
  
• Simple Matlab code to mix students into random groups
  
• Quiz 5 solutions here
  
• HW VI due 2.27.10 (at 5:59 PM) - (posted 2/19) The sixth HW assignment will be up this afternoon. It deals further with Taylor series and its various applications. Useful book chapters are HH 10.1-10.3. Note that you are also asked to do a problem from your ODE book. You may well be asked to turn this in, so make sure to do it! Lastly, don't forget about the online discussion forum (i.e., 'Reading Assignment 1'). The sooner you participate, the better!!
  
• Quiz 4 solutions here
  
  
• Reading Assignment 1 - (posted 2/15) This is the first reading/discussion assignment for 250. The basic gist is as follows: you will read a short article, then contribute to an online class discussion via D2L. The overall purpose is to start drawing parallels between the mathematical content we are exploring in 250 and how it relates to well known scientific problems. The article for discussion deals with historical debate stemming from astronomy.
  
  + The article can be accessed via this Physics Today site.
  
  + The initial seed for discussion will be the following question: how do the various types of mathematics we discuss in class relate the content raised in the article? For example, how do differential equations connect to describing the motion of a planet? Or how mathematically does the 'attraction law' fit into the picture? You are free to pursue threads stemming from this core topic (and even encouraged to do such!), but ideally discussions should connect back to this main point of discussion.
  
  + Your grade for the assignment will contribute towards 'Reading & Discussion Assignments' portion of your final grade (as described on the course website). You will be graded as follows: you are expected to add a minimum of two substantial posts to the forum. Each post will be accessed on a scale of 1 to 3: 1-little content/contribution to the post, 2-adds in a point to the flow of the discussion, 3-makes a significant addition or insight to the discussion. For most, this grading scheme will be somewhat moot: people tend to get engaged and a fruitful discussion flows fairly naturally (thus such participants will readily receive full credit).
  
  + You should be able to access the forum via logging onto D2L. If you have trouble, contact me (CB). The forum will close for additional posts on 2/27 at 5:59 PM, so consider that the official DUE DATE.
  
  
• HW V due 2.20.10 (at 5:59 PM) - (posted 2/12) The fifth HW assignment will be up this afternoon. It deals further with series convergence, as well as determining the radius of convergence. Furthermore, it also has preliminary problems dealing with Taylor series 'approximations' (called Taylor polynomials). Useful book chapters are HH 9.4, 9.5 and 10.1.
  
  
• HW IV due 2.13.10 (at 5:59 PM) - (posted 2/2) The fourth HW assignment will be available noon on Friday 2/5. The assignment is fairly short, given your test earlier in the week on Monday 2/8. This HW deals further with infinite series and convergence. Useful book chapters are HH 9.4 and 9.5.
  
  
• Quiz 3 solutions here
  
  
• Exam I (02.08.10) - (posted 2/2) Though it has come quickly, the first exam is almost upon us. The exam will be on MONDAY morning (2/8). I will not be around, but Prof. Kevin Lin will be there to proctor it. Note that the drop date is the following day (Tuesday, 2/9). Unfortunately, I will be out of town though and thus unable to grade the exams and get them back to you until later in the week. The following is a list of topics covered over the past few weeks that will form the basis for the first exam. Note that this list is not exhaustive (i.e., anything we covered in class or on the HW is fair game!). Furthermore, this material builds off things we covered in 250A (e.g., methods for solving integrals, etc.), though the exam will focus entirely on concepts we've dealt with this semester. So use below as a guide only (also including relevant book chapters to look at for reference, HH for the calculus book and LL for the ODE book):
       - Evaluating Improper Integrals   HH 7.7
       - Convergence of Improper Integrals (e.g., comparison test)   HH 7.8
       - Sequences, Infinite Series   HH 9.1, 9.2
       - Geometric Series   HH 9.2
       - Convergence of Infinite Series   HH 9.3, 9.4
  NOTE: There are lots of good 'practice' problems at the end of Chs.7 and 9 (in the review section).
  
  
• HW III due 2.06.10 (at 5:59 PM) - (posted 1/29) The third HW assignment is now posted on WebAssign. It deals with geometric series and various means for testing for convergence in the case of infinite series. Helpful book chapters are HH 9.2-9.4. Note that for some of the latter questions asking whether a given series converges or not, you only have one shot. So choose carefully!
  
• Quiz 2 solutions here
  
• HW II due 1.30.10 (at 5:59 PM) - (posted 1/22) The second HW assignment is now posted on WebAssign. It further covers improper integrals stemming from HH 7.7 as well as issues dealing with limits (see HH 1.8).
  
• Quiz 1 solutions here
  
• HW I due 1.23.10 (at 5:59 PM) - (posted 1/17) The first HW assignment is now posted on WebAssign. It is relatively short (10 problems) and covers improper integrals stemming from HH 7.7. Due to the late posting, it will be due Saturday rather than Friday. Note that we will still have a quiz on this material Friday morning (1/22).