Math 115-001, Spring 2007

Announcements:

May 5. The final exam was checked. The results and the final grades are posted now on the gradebook. It seems that our average is similar to the average of the second section but our variance is smaller, in particular we got a little bit more B's and less A's and C'. You can see your exams at my office on Monday, 2-4 pm, and I'll submit the results to the secretary on Monday evening. Have a great summer!
May 1. I'll hold a review on Wednesday, May 2, 6-8pm, at DRL 4C8. We'll discuss the first part of the course (that was covered by first midterm).
May 1. If a normal distribution table shows up in the final exam, then it will be a table of usual Phi(t) given by pr(X<=t) (unlike many other final exams, where a table of Phi(t)-1/2=pr(0<=X<=t) was given) !
April 27. The final exam will take place on Thursday, May 3, 12-2pm at Levine Hall, room 101.
April 26. I'll hold a review session on Friday, April 27, 5-7 pm, at DRL 4E19. I'll discuss the second part of the course (practiced by problems from past final exams).
April 22. The problems on variance and expectation were confused with problems from other topics. This mistake is fixed now. Please let me know if you find other mistakes.
April 21. The list of topics for the final is here!!!
April 21. The note on random variable was completed by adding two sections on Poisson and exponential distributions.
April 19. This Sunday there will be a Sunday review at 3-5pm in DRL A5. It covers the whole course. Please prepare your questions.
April 18. I posted a review on random variables (without sections on Poisson and exponential distributions).
April 14. I added a new section to this webpage. It contains problems from old final exams which I recommend to solve. Each day I'll add a new theme (or two themes) and give a list of relevant problems. See section "Problems by themes and old final exams" below.
April 13. The grades for the second midterm are now available on the gradebook. Also, I posted a handwritten solution on the bottom of this page.
April 9. A review session before the midterm will be held on Tuesday, April 10, 5:30-6:30 pm at DRL 4C6. There is no homework this week.
April 4. I will be replaced by Professor Tong Liu on Friday.
March 30. I posted review of linear algebra and discrete probability parts of the course.
March 26. Finally, the second midterm will be held on Wednesday, April 11 !!
March 6. I posted HW7. It contains problems on the first chapter of [P] and will be used in the next quiz (the first week after the break). I'll be replaced by professors Ward and van Erp on Monday, 3.12 and Wednesday, 3.14. They will start the second chapter of [P].
March 6. The results of the make up were uploaded on the blackboard.
February 23. The make up will be held on Thursday, March 1, 6pm at DRL 4C6. Each student whose grade is below 70 can participate. You can neither reduce your grade nor get a grade above 70 (except the students who missed the midterm). Each student wishing to participate should notify me by an e-mail.
February 14. The first midterm will be held in class on February 16. It covers all multivariable calculus (7 problems) and linear algebra, sections 2.1-2.5 (3 problems). The duration is 50 minutes, please do not be late. Good luck.
February 13. I will hold a review session on Thursday, February 15, 6-7:30pm. The location is DRL building, room 4C4. The main theme of the review is linear algebra, sections 2.1-2.5. Please prepare your questions.
February 7. I changed the time of my Wednesday's office hour. The new time is 6-7pm. I hope it will allow to avoid time conflicts with your classes.
February 7. Chenxu and me decided to assign and collect homework on Fridays. It is done to equalize Tuesday's and Thursday's recitation groups. You can submit the homework to me on Friday's lecture or to Chenxu on his Friday's office hour 4-5pm. After 5pm homework will not be accepted. (Also, you can submit the homework on any recitation or lecture before Friday.)
January 31. A part of the last problem of homework 3 was marked as a bonus problem: it turned out that the system of equations obtained in that problem is too messy. I consider writing down the correct system of equations as the main part of Lagrange multipliers method (and only reasonable systems of equations may appear in exams!), so the correct system of equations will provide you with the maximal grade, and you'll get bonus points for the full solution.
January 22. The last problem of homework 2 was marked as a bonus problem.

General Info:

Instructor:

Michael Temkin

E-mail: temkin

math.upenn.edu

Office: DRL 4N61 (215 - 573 - 1029)

Office Hours: Mon 2-4pm, Wed 6-7pm, or by appointment

Lectures: MWF 11-12 TOWN

309

Teaching Assistant:

Chenxu He

E-mail: hech

math.upenn.edu

Office: DRL 4N21 (215 - 898 - 5043)

Office Hours: Tuesday 12:00noon - 1:00pm, Friday 4:00pm - 5:00pm

Recitations: DRLB 4E9

115-201: T 8:30-9:30
115-202: T 9:30-10:30
115-203: R 8:30-9:30
115-204: R 9:30-10:30

Texts:

[C] Thomas/Finney, Calculus, 9th (or Alternate) Edition.

[P] DeGroot/Schervish Probability and Statistics, 3rd ed

[F] Lial, Greenwell, Ritchey Finite Mathematics, 7th ed.

Maple/Calculus Lab Manual for Math 103/104/114/115.

Evaluation:

TA's grade (homework & quizzes) 20%

Midterm I; Friday, February 16 20%

Midterm II; Friday, April 13 20% The final date is Wednesday, April 11 !!

Final Exam; Thursday, May 3rd, 12-2 pm 40%

The final exam will take place at Levine Hall, room 101 !!

All grades will be posted on the Blackboard. The midterm exams will be held at class, their dates are tentative.

Course description:

Here is math 115 homepage of Math Department. It contains syllabus, core problems, old final exams, etc.

The course consists of three parts: (1) Multivariable functions and (2) Matrixes (approximately half a course), (3) Probability. Maple is an optional tool which can turn out to be helpful in your studies. It can also be used in lecture demonstrations.

Exams:

All exams are without books and calculators. You may bring one handwritten cheating sheet of size 11.5"x8" (both sides).

Homework:

The exercises are assigned and collected on Fridays, but you can submit a homework on any lecture or recitation before the due date. Each homework assignment will be posted on this webpage. Your are allowed, and even encouraged, to work in small groups, but each student should write down the solution separately. Identically written (or photocopied) homework may be declined!

Homework Assignments:

Bonus problems are marked with *, they are not obligatory.

Homework 1. Due January 17. Ex. 12.1/7,8; 12.2/12,14,42; 12.3/24,44,52,57.
Homework 2. Due January 24. Ex. 12.4/6,11,20 a),26 ; 12.5/6,13,17,41*.
Homework 3. Due January 31. Ex. 12.7/22,28,32,36 (find tangent planes and normal directions); 12.8/10,17,32,36; 12.9/17,22,24,30.
Homework 4. Due February 7. Ex. 13.1/4,6,13,25,30,38; Chapter 12 Review (pp. 994-998) 26,31,62,69,73,82.
Homework 5. Due February 16. Ex. 2.1/24; 2.2/13,20,26,30; 2.3/15,24,26;2.4/17,19,26,31,38.
Homework 6. Due February 23. Ex. [F] 2.5/19,20,35,36; 2.6/1,5,7; 1.3/5,6 (you can use a calculator in the last two problems).
February 23: no homework this week. Prepare to the quiz (and makeup). Next homework will be assigned on March 2.
Homework 7. Due March 16. Ex. [P] 1.7/3,5,7; 1.8/2,3,7,10,12,15; 1.9/4,9; 1.10/5,6.
Homework 8. Due March 23. Ex. [P] 2.1/2,3,6,9,11; 2.2/7,13,14,15; 2.3/1,9,10.
Homework 9. Due March 30. Ex. [P] 2.4/2a,b,12, also find long term run probabilities in 2 and 12; 3.1/2,5; 3.2/4,5,7.
Homework 10. Due April 6. Ex. [P] 3.4/1ab,5abc; 4.1/2,3,9; 4.2/2,3,6,9,10.
There is no homework this week, but I recommend to solve the following problems from old final exams. Fall 2004: 10,12,15,16; Spring 2005: 6,9,10,15,18; Fall 2005: 1,6,14,15,18; Spring 2006: 6,7,8,9,10,11,18. The suggested time is 3-6 minutes per problem (6 minutes can be spent only for the most computational problems, e.g. bivariate distributions or input/output models).

Lectures:

Each lecture's title is followed by relevant core problems. Make sure you can solve them.

January 8. Multivariable functions, [C] 12.1. Ex. 5,8,13-19.
January 10. Limits and continuity, [C] 12.2. Ex. 11,13,16,35.
January 12. Partial derivatives, [C] 12.3. Ex. 5,19,30,47,57,63,65.
January 17. Linearization and differentials, [C] 12.4. Ex. 5,11,20,23,24.
January 19. The chain rule with one independent variable, [C] 12.5. Ex 3,40.
January 22. The chain rule with two independent variables, [C] 12.5; gradients and tangent planes, [C] 12.7. Ex. 12.5/8,17,41; 12.7/2,3,27,31.
January 24. Gradients and directional derivatives, [C] 12.7; Extreme values and saddle points, [C], 12.8. Ex. 12.7/17,18; 12.8/6,11,17,29.
January 26. Absolute extrema, [C] 12.8. Ex. 36,39,42.
January 29. Lagrange multipliers, [C] 12.9. Ex. 17,21,23,32,46* (read Example 5 in the text before solving it).
January 31. Double integrals and Fubini theorem, [C] 13.1. Ex. 5,8,15,45,53,54.
February 2. Double integrals, finding the bounds of integration and switching the order of integration, [C] 13.1. Ex. 21,23,26,29,33.
February 5. Review of multivariable calculus, [C] 12.1,2,3,4,5,7,8,9;13.1. Brief review notes which were shared at class can also be found here. Repeat the material using the text, the review notes and your lecture notes. Make sure that you remember how to solve all previous homework. Prepare your cheating sheet with basic formulas. Make sure that you know how to solve the following practice exercises using the cheating sheet only: pp. 994-998/ 3,21,26,31,39,57,59,71,81,85; and 13.1/21,23,25,31,36,39. Exploit my or TA's office hours if you experience difficulties with the exercises.
February 7. Systems of linear equations and the echelon method, [F] 2.1. Ex. 23,25,29. Read section 2.1 and look through section 2.2.
February 9. Gauss-Jordan elimination, [F] 2.2. Ex. 17,24,27,28,33,39,40. Read section 2.2 and look through sections 2.3 and 2.4. An example of Gauss-Jordan elimination is here. This example also illustrates how systems of linear equations appear in other sciences.
February 12. Addition and multiplication of matrices, [F] 2.3 and 2.4. Ex.
February 14. Inverse matrices, [F] 2.5. Ex. 11,13,15,17,19,22,23. Read section 2.5. Prepare to the midterm.
February 16. Midterm I. Look through section 2.6.
February 19. Open input-output models, [F] 2.6. Ex. 1,5,17. Read section 2.6 and look through section 1.3.
February 21. Closed input-output models, [F] 2.6; The least squares line, [F] 1.3. Ex. 8,10,11,17. Read section 1.3 and look through section [P] 1.5 (sections 1.1-1.4 of [P] may be useful too).
February 23. Set theory, [P] 1.4; The definition of probability, [P] 1.5. Ex. 1.4/3,4,5,6; 1.5/3,7,8,9,10. Read sections 1.4,1.5 and look through sections 1.6,1.7,1.8.
February 26. Finite sample spaces, [P] 1.6; Counting methods [P] 1.7; Combinatorial methods [P] 1.8. Ex. 1.6/1-4; 1.7/1,3,4,5,7,8. Read sections 1.6,1.7,1.8 and look through section 1.9.
February 28. Combinatorial methods, [P] 1.8. Ex. 2,3,4,7,17,18. Read section 1.8 once again, read section 1.9, look through section 1.10.
March 2. Multinomial coefficients, [P] 1.9; Probability of union of events [P] 1.9. Ex. 1.9/1,3,6,8; 1.10/1,4,5,6,9.
March 12. Conditional probability, [P] 2.1. Ex. 6.9.10. Read section 2.1 and look through section 2.2.
March 14. Independent events, [P] 2.2. Ex. 7,8,9,10,12,16,19. Read section 2.2 and look through section 2.3.
March 16. Bayes' theorem, [P] 2.3. Ex. 1,2,10,11,12. Read section 2.3.
March 19. Markov's chains, [P] 2.4. Ex. 2,3,11,12; also find the stable probabilities in these exercises. Read section 2.4 and repeat the whole discrete probability part of the course, [P], chapters 1 and 2.
March 21. Random variables and discrete distribution, [P] 3.1. Ex. 2,3,4,8.
March 23. Continuous distribution, [P] 3.2. Ex. 2-8. Read sections 3.1 and 3.2 and look through section 3.3.
March 26. The distribution function, [P] 3.3. Ex. 3-8. Read section 3.3 and look through section 3.4. Repeat double integrals towards the next lecture !!
March 28. Bivariate distribution, [P] 3.4. Ex: see the first three problems in the end of math115 syllabus. Read section 3.4 and look through section 3.1.
March 30. Expectation, [P] 4.1. Ex. 1,2,4,5,9. Read sections 4.1 and 4.2.
April 2. Properties of expectations, [P] 4.2. Ex. 6,7,8,10. Look through section 4.3.
April 4. Variance, [P] 4.3. Ex. 1,2,3,6,7,9. Repeat the material of chapter 4.
April 6. The Poisson distribution, [P] 5.4. Ex. 2,3,6.
April 9. The normal distribution, [P] 5.6. Ex. 3,7,9,10.
April 11. Midterm II.
April 13. The normal distribution, [P] 5.6; the exponential distribution, [P] 5.9. Ex. 8,9,13,14.
April 16. Repetition on I/O models, Markov's chains, composition and permutation numbers.
April 18. Repetition on Bayess' theorem, double integrals, bivariate distributions and expectations.

Problems by themes and old final exams:

This section contains themes we have studied, references to text and to review notes and few relevant problems from final exams. Also, I'll try to mention standard mistakes. I'll post one or two themes each day, and Chenxu will post a scanned solution of one problem a day after. The following abbreviations are used: texts [C], [F] and [P] as usual; notes [Cal], [LinProb] and [RandVar] ; final exams [F6] (Fall 2006), [S6] (Spring 2006), [M5] (Makeup 2005), [F5] (Fall 2005), [S5] (Spring 2005), [F4] (Fall 2004) and [S4] (Spring 2004).

April 22. Exponential distribution; [P] 5.9 and [RandVar] 11. Problems: [F6] 10; [S6] 13; [F5] 5; [S5] 12; [S4] 14.
April 21. Poisson distribution; [P] 5.4 and [RandVar] 10. Problems: [F6] 12; [S6] 14; [M5] 2; [F5] 2; [S5] 16; [F4] 17; [S4] 17.
April 20. Binomial distribution; [P] 5.2 and [RandVar] 8. Problems: [F6] 13.
April 20. Normal distribution; [P] 5.6 and [RandVar] 9. Problems: [F6] 18; [S6] 19; [M5] 17; [F5] 17; [S5] 17; [F4] 8; [S4] 19.
April 19. Variance; [P] 4.3 and and [RandVar] 7. Problems: [F6] 9; [S6] 9; [M5] 1; [F5] 1; [S5] 10; [F4] 18. Mistakes: wrong formulas, e.g. wrong computation of E(X^2).
April 19. Expected value; [P] 4.1,4.2 and [RandVar] 6. Problems: [M5] 6; [F5] 6; [S5] 7; [F4] 11; [S4] 10. Midterm, problem 7. Mistakes: wrong formula (especially when the expectation of h(X) rather than of X itself should be found). Another type of mistakes in the midterm was caused by the fact that a distribution function (d.f.) was given; it was often confused with its derivative p.d.f. (the probability density function).
Double integrals and bivariate distributions is one of the most difficult themes. It must be practiced a lot.
April 18. Bivariate distribution; [P] 3.4 and [RandVar] 5. Problems: [S6] 10; [S5] 9; [F4] 15; [S4] 13. Midterm, problem 8. Same mistakes as in the case of double integrals, and wrong formula for joint p.d.f of independent random variables. Solution can be found on Chenxu's webpage.
April 18. Double and repeated integrals; [C] 13.1 and [Cal] 12. Problems: [F6] 5; [S6] 4,5; [M5] 11; [F5] 11; [S5] 4,5; [F4] 14; [S4] 5. Midterm 1, problems 6,7. There are various mistakes including wrong graphs, wrong bounds of integration and computational mistakes. Solution can be found on Chenxu's webpage.
April 17. Least squares; [F] 1.3 and [LinProb] 1. Problems: [S6] 12; [F5] 11. Solution can be found on Chenxu's webpage.
April 17. Bayess' theorem; [P] 2.3 and [LinProb] 11. Problems: [F6] 8; [S6] 6; [M5] 13; [F5] 14; [F4] 16; [S4] 7. Midterm, problem 5. Mistakes: wrong probabilities (for example pr(A)=pr(B)=pr(C)=1/3 in the midterm), computational mistakes. Solution can be found on Chenxu's webpage.
April 16. Union of events; [P] 1.10 and [LinProb] 9. Problems: [M5] 16; [F5] 16. Midterm, problem 3. Solution can be found on Chenxu's webpage.
April 16. Multinomial numbers; [P] 1.9 and [LinProb] 9. Problems: [S6] 11. Midterm, problem 4. Solution can be found on Chenxu's webpage.
April 16. Composition numbers; [P] 1.8 and [LinProb] 8. Problems: [F6] 7,11; [S6] 7,8; [F4] 10; [S4] 8. Midterm, problems 1,2. Solution can be found on Chenxu's webpage.
April 15. Counting, permutation and composition numbers; [P] 1.5-1.8 and [LinProb] 6 and 7. Problems: [F6] 6; [M5] 15; [F5] 15; [S5] 6,8; [F4] 9; [S4] 6,9. Midterm, problems 1,2. Solution can be found on Chenxu's webpage.
April 14. Markov's chains; [P] 2.4 and [LinProb] 5. Problems: [S6] 18; [S5] 15; [F4] 6; [S4] 16. Midterm, problem 10. Mistakes: wrong transition matrix (confused rows and columns); remember that the sums in each column are equal to one. Solution can be found on Chenxu's webpage.
April 14. Input/output models; [F] 2.6 and [LinProb] 12. Problems: [M5] 5,18; [F5] 18; [S5] 18; [F4] 12; [S4] 18. Midterm, problem 9. Mistakes: wrong I/O matrix (confused rows and columns). Solution can be found on Chenxu's webpage.

Notes:

February 5. A review of the multivariable calculus part of the course is here.
February 12. You can find here brief notes where a chemical reaction is balanced by solving a system of linear equations by Gauss-Jordan method.
March 30. A review of the linear algebra and discrete probability parts of the course is here.
April 18. A review on random variables is here.

Solutions:

February 22. A handwritten solution of the first midterm is here.
March 6. A handwritten solution of the make up to the first midterm is here.
April 13. A handwritten solution of the second midterm is here.