Math U142 class notes, Spring 2005

Prof. A. Iarrobino


Section meeting MWTh at 10:30 AM in 235 Forsyth,


Look here for specific assignments and brief comments on what we did (or will do) in class. TBA = "to be announced"


Syllabus (goals, assignments) for all sections Math U142 syllabus (pdf file)

Syllabus: Information sheet (grading, etc. for Prof. Iarrobino's sections) Math U142 Information/syllabus (pdf file)

Course home page (you probably already know this information): Math U142 home page (html)

Course outline (broad strokes) Course Outline



First day of classes: Wednesday January 5 Went over syllabus - class information. Differentials, linear approximation (Section 6.5). Begin substitution method of integration (Section 7.2).

Homework due Monday: Section 6.5 p. 361 #1,3,5,19-21,26.

Thursday January 6. Questions about linear approximation, #26 p. 362.
Substitution method. Begun Worksheet #1A on substitution method, #1A-C. Two ways to handle the algebra. Substitution method with definite integral (change of limits of integration).
 Worksheet #1A (pdf)

Homework due Monday Jan. 10: Section 7.2 Choose about half of p. 387 #2-34 to do.
Worksheet 1A #1D (transform to u-integral with new limits), 1E, 2A (transform to u-integral).

Week of January 10 Continue Chapter 7, integral (much of this reviews last portion of Math U142).
Quiz #1 (Thursday)

Monday January 10. Questions mainly from section 7.2 and the worksheet #1A, on substitution method. Passed out a worksheet on linear approximation (from M. Nathanson).

Homework for Wed.: Complete problems on substitution method in WS #1A (carrying as far as indicated in class), and the HW from above in 7.2.
Complete the worksheet on linear approximation, to prepare for quiz 1.
Review section 7.1 (be able to do problems). Read Section 7.3 in preparation for Wednesday.

Wednesday January 12: The integral as area, left, right, midpoint, and trapezoid sums for the integral (section 7.3).

Homework for Thursday Jan. 13: Begin Section 7.3 HW. p. 398 #3,4,6,9,13,16,17.

Thursday January 12: Questions. Quiz #1 on topics of past week (at least 35 minutes)

Homework for Wednesday January 19. Complete Section 7.3 HW above; Also Section 7.3 #23-26,29,32,33.
Read Section 7.4. Do p. 409 #1-5, 53.

Week of January 17 Continue with integration: integral as change in F equal to area under F'(x) (section 7.3), fundamental theorem (Section 7.4), trig functions (Section 7.5)

Monday January 17 (Martin Luther King holiday, NU closed).

Wednesday January 19: Passed back Quiz 1, and solutions, went over #1b, 2c,3b. Questions on Section 7.3: we went over #23,24, and 29 p. 399-401.
Use of F(b)-F(a) = integral of F'(x) dx from a to b , which is the signed area between the y=F'(x) curve and the x-axis.
We went over the Fundamental Theorem: (see p. 403).

Homework to begin for Thursday, complete Monday Jan. 24.
Complete Section 7.3 HW above
Section 7.4 p. 409: #1-23 (do a third of these, as every third problem); 41-43, 46, half of 53-62.
I plan to check HW Thursday or Monday.

Thursday January 20 (plan): Question on Fundamental Theorem (Section 7.4), Integrals of Trig functions (Section 7.5).

Homework for Monday Jan. 24
Complete Section 7.4 homework above, also #65,66,p. 413.
Section 7.5 p. 418 #1-15 odd, 25-26, 31,32,35.

Monday Jan. 24 NU closed (weather)

Homework for Wednesday Jan. 26 (in addition to that for Monday): Read Section 7.6 (area between curves), Begin HW p. 424 #1-9,27,29.

Note on class discussion of p. 412 #64. The class had good intuition in understanding the integral in 64a. However, the function n(x), the integrand, must then be the limit as dx goes to zero, of the following quotient: numerator/denominator
Numerator: the number of cells with percent probability of division in the next hour between x and x+dx percent
Denominator: dx percent.
This is somewhat different than n(x) as defined in the problem: in fact, for most probability distributions, there will be zero cells having exactly x percent probability of division in the next hour. For more info see Chap. 13 (we will be discussing this later)

Wednesday Jan. 26 Check HW due Monday, Questions, Trig integrals, area between curves.

Homework for Thursday Jan. 27: Finish Section 7.6 HW above p. 424 #1-9,27,29.
Also same section #35,36 (new)

Thursday Jan. 27 (plan) Questions, applications, integrals using tables (WS #1B pass out, tables in text p. A-4).
Approximating an integral: trapezoid and Simpson's rules (Section 8.1).

Homework for Monday Jan. 31: Do Syllabus assigment for Chapter 7 Summary and review, p. 428
Begin Section 8.1: p. 445, 1,5,7,24,25,27.

Sample Quiz 2 (pdf)
Sample Quiz 2, solutions to #1-3 (pdf)

Monday Jan. 31. Questions, Quiz #2.

Wednesday Feb 2. Went over Quiz #2 solutions, passed out solutions. Simpson's rule (section 8.1), Began volumes.

Thursday Feb 3. Volumes using disks, rings (for volume between two rotated curves): Section 8.2 and Worksheet #2. Began integration by parts (section 8.2)

Homework for Monday Feb. 7:
Section 8.1 (Simpson's rule): p. 443 #1, 5-7, 15-16,24,27, 31,32,34,35.
Read Section 8.2 (integration by parts), begin p. 454 #1-3.
Section 8.3 (Volumes) p. 461 #1-5 odd, 23,24,27,38-39.
Worksheet #2: Finish problems

Please be safe on Sunday.

Monday Feb. 7. Check HW. Questions on volume. Integration by parts.

Homework for Wednesday Section 8.2 p. 454 #1-11 odd, 21,23,37,38, 40-44.

Wednesday Feb. 9. Check HW due Monday (continued). Questions on WS #2: problems #3A,3B (much of class on #3B, written as V_1+V_2-V_3, involving three integrals. Began improper integrals (Section 8.4).

HW for Thursday Section 8.4 #1-3. (more will be assigned for Monday).

Thursday Feb. 10. Questions. Improper integrals. Half quiz 3 (Chapter 8).

HW for Monday Feb 14
- Section 8.4 p. 467 #1-8 (enough to get practice), 27,33, 37-38 (especially, we began 37 in class), 44-45.
Chapter 8 summary. p. 469 #6-9, 11-15, 27-29, some from 33-40 and 43-44, Applications: 45-49 (but in #46b, just be able to set up Simpson(20) sum).
Please read section 9.1 (a first read).
Practice Exam 1 from Spring 04 (except #3D), bring in questions Monday.

The following will give you practice for Exam 1. We will discuss partial derivatives (section 9.2) on Monday Feb 14.
Math U142 Spring 04 Exam 1 (pdf file)
(Note the diagram is missing from #3D of two velocity curves: the function v_B=2.5x; the function v_B begins at (0,0), line to (5,10), then line to (6,30), then a parabolic arc concave down but positive slope, to (8,35).).
Math U142 Spring 04 Exam 1 solutions (pdf file) [Adjusted to include previously omitted answers]

Monday Feb. 14 Half Quiz 3 back, with solutions: we went over solutions, and also average value (from section 8.3, quiz problem #4).
Went over class feedback and response.
Valentine day special: Working with many variables. We began section 9.2 partial derivatives (p. 490-492) and will return to 9.1 later. Interpretation of partial derivative with respect to x, as slope in x-direction along z=f(x,y) at a point.
Also, passed out practice problems from M. Nathanson for Exam 1. Solutions will be posted when ready.

Homework for Wednesday Feb 16 Section 9.2 p. 495 1-8, 23, 52 (last two added on web).

Extra Office Hour Tuesday Feb. 15, 1-2:30 PM

Wednesday Feb. 14 Practice for exam, review.

NOTE I have posted an adjusted pdf file of solutions to Exam 1 in 2004, that include previously omitted problems (it was a problem of page size).

Monday Feb 21: no class, NU holiday

Homework for Wednesday Feb. 23 Please read Sections 9.1-9.2, and, if you wish to prepare for Wed class, also Section 9.3.
Begin p. 484 (section 9.1), #1, 19. We will discuss this section and p. 485 #22-27 Wednesday
Section 9.2 p. 495: further HW: #1-8,18-25, 52, also try 35-36, and 49 (see Example 4 p. 492 for #49)

Wednesday Feb. 23 Exam 1 back with solutions. Section 9.3 Local max-min. Also, related material in Chapter 9.2, as in #35-36 p. 495. Test for relative extrema (max,min,saddle): see p. 502 of text. Students did Section 9.3 p. 506 #1,4,21 in class.

Homework for Thursday Feb. 24 Section 9.3 p. 506 #1-8, 13 (added), 21-23.

Thursday Feb. 24 (Plan) Section 9.3, continued: students did #13,22 in class. Section 9.4: approximation using the differential (Section 6.5 is the analogue for functions of one variable). Students did p. 512 #9, and we did #15 in class.

HW for Monday March 7
Section 9.3. Complete above HW, also #32-33 (in #33, I would suggest to fix alpha, so f is a function of two variables, then there are four critical points.
Section 9.4. p. 512 #1-3, 9-111, 15-18, 20-25.

Week of March 7 Questions on Section 9.4, Section 9.5 double integrals. If time, begin Section 11.1 on differential equations.

Monday March 7 Questions on Section 9.4 (p. 513 #2,15,17,24, part of #25); Section 9.5 multiple integrals and volume.
Principal: signed volume under z=f(x,y) and above z=0, above a specified region R, is the multiple integral over R of f(x,y).

HW for Wednesday, March 9 Section 9.5 p. 524 #1-7 odd, 13-17 odd, 23,25 (enough of these to have idea)
#33-37, 41-44, 53-55.

Wednesday March 8 Questions on multiple integration. We did in class p. 524 #5, 15 (horizontal strips, also, reversing the order to use vertical strips),34,42,55 (vertical strips; also we set up integral in reverse order, using horizontal strips).
We practiced giving a graph of the region of integration, and using that graph to change the order of integration (#55).

Homework for Thursday, March 10#54, 57-61; 64-65; also, in #54, 59, 60 set up integrals in reverse order for these.

Thursday, March 10 Questions, then Quiz 4 on Chapter 9 (see below for main topics)

Week of March 14 Begin Chapter 11, to be followed by Chapter 13.

Monday March 14 Passed back Quiz 4 and went over it. Passed out worksheet on multiple integration, and class worked on second page (old quiz).
Began Section 11.1: differential equations, separation method of solving (some of) them.

Homework for Wednesday March 16
Finish worksheet as needed
Chapter 11.1. p. 611 #5-9, 19,20; begin 25, 30, 39. (latter added on web, so part of Monday March 21 assignment)

Wednesday March 16 Section 11.1 differential equations, separation of variables. Students put solutions to #5-9, 19-20. We also solved #30, and on GC noted shape of solution curve.

Homework for Thursday March 17: graph on GC y=(-1/((2/3)x^{3/2)-100), and compare with solution to #30 IVP.
Also, begin HW for Monday, bring in questions.

Thursday March 17 Continue with separation of variables.
Newton's law of cooling: y'=k(N-y) has solution y=N-Ae^{-kt}, and equilibrium y'=0 when y=N. We interpreted this (see Section 11.1 #38 and Section 11.2 #31-33, note that c there is (-A) here.)
Logistic equation (p. 609-610): y'=ky(1-y/N). Solution y=N/(1+be^{-kt}), and two equilibria y'=0 when y=0 or y=N.
We began Section 11.2, first order linear diff. equations, integrating factor.

Old HW for Monday March 21
p. 611 #1-15 as needed to understand process (#5-9 already assigned).
p. 611 #19-23, 30
p. 611 #38-42 (use text p. 608-610, where solutions to ODE y'=k(N-y) and y'=Ky(1-y/N) are given, see notes to Thursday, above)

New Homework for Monday March 21
p. 621 #1-7 (if needed), 15-17, 19, 23-25, 31, 34-35.

Monday March 21 Checked Homework above. We continued practice solving first order equations using integrating factor (Section 11.2): we solved (on board) p. 621 #5,6,16; also #1 from Worksheet #4 (passed out). Passed out also Exam 2 from Math U142 in Spring, 2004.

Homework for Wednesday Complete HW Section 11.2.
Exam 2 from Spring 2004 p. 5-6 except last problem from 11.3.
WS #4 (continue)

Wednesday March 23 Questions: WS 4, problems #1,2. Intro to direction field, using y'=x^2+y^2. Section 11.3, Euler's method. First problem, Estimate y(0.6) for y'=x^2+y^2, with y(0)=1, and interval h=0.2. Second problem: we began p. 629 #31.
Method: We make a table of values with first column x, second y, and third column y', we obtain y(new) = y(old) +{y'(old)}( h), and we obtain y'(new) from the differential equation.

Homework for Thursday p. 628 #1,3,17,18, 31.
Prepare for Quiz 5. See rest of WS #4, and Exam 2 from Spring 04 p. 5-6.

Office hour: Wednesday March 23 12-12:30, 2-3, 4:15-5:15 (longer if needed).

IF YOU HAVE Questions on Euler's method, feel free to come to the other section of Math U142 at 9:15 AM Thursday March 24 at Rm 102 Kariotis: I will be substituting, and their homework for Thursday was also on Euler's method. (Then come to the regular class)

Thursday March 24 Questions (up to 20 minutes). Quiz 5 (Chapter 11.1-11.3). Hand out Sample Problems for Exam 2.

   Quiz 5:March 24 on Chapter 11 (differential equations). Four problems, one a work problem on NLC or Logistic equation (see Worksheet 4 above), one separation of variables ODE, a first order ODE, and an Euler's method approximation.

Week of March 28 Begin Section 11.6 Applications of ODE. Practice for Exam 2, Exam 2 on Thursday March 31. Note: we skip Sections 11.4,11.5: these involve systems of two linear differential equations.

Monday March 28 Section 11.6 (Applicatiosn of ODE) p. 650 we solved #3,10, and set up #11.
Passed out solutions to Sample Problems for Exam 2.

Homework for Wednesday Section 11.6 p. 650: #1-4,6-9, 11-16.
Sample Problems for Exam 2 (answers passed out Monday)
Please go over your Quiz 5 and solutions (good practice for Exam 2).
Optional: For additional practice: Choose from Review problems, p. 652 #5-12,13-20,25-27,39 (Euler), 47-48, 50-52,59-60.

Wednesday March 30 Section 11.6, continued (questions), practive for Exam 2.

Thursday March 31 Exam 2 (see below for sections covered).

Here are downloadable pdf files of the sample problems and solutions for Exam 2; these were handed out Thursday March 24 and again on Monday March 28 (a few students missed both Thursday and Monday).
Sample problems for Exam 2 Solutions to the sample problems

Homework for Monday April 4 Read Section 13.1.

Week of April 4 Chapter 13 Probability Distributions.

Note TIME CHANGE:
Spring forward: advance clocks one hour on Sunday April 3! To check correct time: phone 617-637-1111 ; (often correct weather forecast: 617-936-1111)

Monday April 4 Begin Section 13.1: Continuous probability models.
Passed out sample problems for final exam.

Wednesday April 6 Questions on 13.1. Begin Section 13.2 Expected value and variance.

Homework on Section 13.2 p. 794: #1,3,5,9-10,11,15,19,21-23,25,27 set up only,30,32.

Thursday April 7 Questions from Section 13.2 : #7,1`5,21,29. Section 13.3: Special Probability Distributions (uniform, exponential, normal, use of z-scores in normal distribution.

Homework for Monday April 11 Complete HW from Section 13.2.
Section 13.3 p. 756, #1,4,7,8,11,13,17,21-23,27,29,31,34b, 36,42.

Week of April 11 Review for final exam. Half quiz 6 on Monday

Monday April 11 Questions. Half quiz (half hour). Evaluation.

Wednesday April 13 (Last class): Questions on sample problems for final exam. Passed back Half Quiz 6 (will count if score is greater than average of other quizzes)

Announcements:

Office Wednesday April 13: 1:45-2:30, 4-4:45.

Unusual Office Hours: Thursday April 14: 3-4:30 PM.

Final Exam grades ready: we will be finishing correcting them Tuesday April 19, grades will be ready Wed. April 20. If there is some issue concerning the final you took Friday, feel free to drop by Tues April 19, 11:30-2 (not an official office hour), or Wed April 20 in the afternoon 2:30-3.

I will be out of town April 21-May 1

Textbooks: Should you wish to sell your text back to the bookstore, you should hold on to it until the Math Dept sends a book request to the store for Fall for U142, provided we use the same book then. Or you should hold the book until Fall 2005 when we send in the book requests for Spring 2006. It is certain that we will teach the course in Spring 2006, and I believe we will use the same text then --- certainly, that is what I would recommend, if I coordinate again.
A section is being considered for Fall 20005, and may use the same text.
You should receive 50% for a text in good condition, if you sell it back just before the semester we use it again.


Future dates of interest:

Final Exam: Friday April 15, 2005, at 3:30 PM, in 322 Hayden. (Data as of April 8, 2005). Two hour exam.
BRING EXTRA CREDIT HW to Final exam: if you have done the homework regularly, you may bring Notebook or HW to final exam, I will look it over and assign an extra credit grade based on amount done, this goes into the (and can help) the Quiz-HW grade.

Course grades (Wednesday April 20) Course grades are now posted outside my office.

Many students in both sections did quite well on the final exam - congratulations!
If you have a question about your course grade, feel free to contact me about it. You may also see your final exam (or I may be able to send you a copy).
However, I will be in Stockholm visiting the KTH math department from Friday April 22, 2005 to May 1; I will not be in phone or effective e-mail contact during that time, and will respond to any communications when I get back.

If you have math-related questions (about courses to take, a math problem, application) feel free to contact me any time, by e-mail is usually the most effective. During the summer I am generally around, but doing research, so I do not maintain regular office hours. However, if you send me an e-mail, and can come to NU, I can arrange to meet you.



Questions or comments: e-mail to iarrobin@neu.edu.  This is the quickest way to reach me, or come by 526 NI, MWTh.

Syllabus (goals, assignments) for all sections Math U142 syllabus (pdf file)

Syllabus: Information sheet (grading, etc. for Prof. Iarrobino's sections) Math U142 Information/syllabus (pdf file)

Course home page (you probably already know this information): Math U142 home page (html)



Math Tutoring Free, at NU, available to Math U141 students.



Links  to other calculus resources on the web:

Visual Calculus (U. Tenn)


Prof. Anthony Iarrobino
Department of Mathematics
Northeastern University
Boston, MA, 02115
Office:526 NI
Phone: (617) 373-5524
Email: iarrobin@neu.edu


Created: January 6, 2005. Last modified: April 20, 2005. URL:http://www.math.neu.edu/~iarrobino/AIMathU142.Spr05.classnotes