-*---* A. Iarrobino, Math U141, Spring 06 Classnotes

Math U141 class notes, Spring 2006

Prof. A. Iarrobino


For Section


meeting MWTh at 9:15 AM in 302 Kariotis.


Look here for specific assignments and brief comments on what we did (or will do) in class. This will be updated at least weekly and usually by 4:30 PM each class day. TBA = "to be announced"


Course-wide syllabus info (Prof. J. Frampton, coordinator, prepared this): Math U141 Course wide info (Syllabus A) (pdf)

Information sheet (grading, etc. for Prof. Iarrobino's sections) Math U141 Information Syllabus B (pdf file)

Syllabus (goals, assignments) for A. Iarrobino section Math U141 Detailed assignments: syllabus C (pdf file)

Course home page (you probably already know this information): Math U141 home page (html)

Course outline (broad strokes) Course Outline



First day of classes: Monday January 9 Went over syllabus, information, took placement quiz; then we reviewed the equation of a line, both slope-intercept y=mx+b form and point slope form. We found points a fraction - as two thirds - along the line segment between two points.

HW for Wednesday: Worksheet #1A, #1.
Also Text Section 1.2 p. 11 #1-3,7,9,22,15,18,20,24,26,27.

Note: Class Pacs will likely be ready at Reprographics (next to book buy-back behind the Campus bookstore) late Friday afternoon (January 13 after 4 PM) or Monday morning.

Wednesday, January 11 (plan) Placements back, questions on Section 1.2. Average rate of change (Section 1.3).

Example of y=4x^2, where y is height of a ballon in feet, x minutes after release. Average slope here has units of feet/minute and is also average velocity of the balloon.
The average slope between P(1,4), Q(3,36) is 16;
The average rate of change between P(1,3) and Q'(1+h, 4(1+h)^2) is 8+4h (after some algebra): this is the slope of the line PQ'
The average rate of change between P(1,3) and (1.01,) is 8.04.

Instantaneous rate of change (velocity here): the limit as h goes to zero of the average rate of change from P to Q is 8 feet per minute, which is the vertical velocity of the balloon at time 1 minute.

Homework for Thursday Jan 12: Worksheet #1A, Problem #2. [This asks for a similar calculation for y=3x^2, to obtain the slope at x=1].
Thursday January 12 Continue average slope, slope at a point (section 2.1). Slope function f'(x), and its meaning. Graph of slope function f'(x), given f(x).

HW for Wednesday January 18
Section 1.3 p. 19: #1-5,9,12,13,23,29.
Section 2.1 p. 103 #1-4,6-9,11,13,23.
Section 2.2 p. 107 #1-7, 9-13.
Quiz 1B Fall 03 (passed out in class).

Note The class packs will be available Tuesday Januray 17 at Reprographics (near the book buy-back window in corridor behind the the campus bookstore: don't wait in the book buyback line, the line for Reprographics is very short. $5. It will have a cover "Math U141 Spring 2006 Class Pack: A. Iarrobino"

Week of January 16 Derivatives, quiz #1 on Thursday.

Monday January 16 Martin Luther King Holiday: no classes.

Wednesday, January 18 Derivatives, continued. Questions on Quiz 1B from Fall 2003 (passed out). Began WS #2B

HW for Thursday Jan . 19:
Please pick up your Class Pack (see Note above)
Quiz #1A, Fall 2003 (p. 35 Class Pac, Solutions p. 73).
Continue WS #2B (p. 10 of Class Pack), #1. (not on Thursday quiz)

Thursday, Jan. 19 Questions (about 20 min), Quiz #1 (about 40 minutes).

HW for Monday January 23:
Read text Section 2.3. (meaning and units of derivative function). Do p. 116 #1-7,9-13, 15,18,23,25,27-30.
Continue WS #2B #1 (we will discuss this further Monday in class)
Also read text p. 135-139) limits (this may help with questions you are having about the limit process).
Begin p. 139 #1,5-7,25,29,
Note: The last two problems #25,29 are just the definition of derivative that we have been doing, using P(x,f(x), Q(x+h,f(x+h)), then finding the slope m_{PQ}, and the limit m_P =f'(x) as h goes to zero).

Week of January 23 Derivatives, higher derivatives, formulas, special functions (exponential, log, trig.

Monday January 23: Quiz #1 back, with solutions. We finished WS #2B #1, then continued with problems in Section 7.3 of text, especially linear approximation along the tangent line (see p. 115 of text).

HW for Wednesday, January 25: continue HW above scheduled for Monday.

Wednesday January 25 : Continue interpretation of derivative (Section 2.3), begin derivative formulas (chapter 3).

HW for Thursday: Section 3.1 of text: syllabus HW
Also WS #3A #3 (from class).

Thursday January 26 : Continue formula derivatives, second derivative, exponential functions and their derivatives.

HW for Monday January 30:
Text Section 1.5 (exponential functions): p. 38 #1-4,6,9,17,18,25-28.
Text Section 3.2 (derivative of exponential function) p. 152 Odd #1-11, 33-35.

Week of January 30: Exponential growth (continued), Derivatives of exponential function, Chain rule, product rule.

Monday January 30 Exponential growth: overview. Modeling exponential growth with y=A_0e^{kx}.
Composite functions (Section 1.8), Chain rule (Section 3.3).

HW for Wednesday Feb 1: Section 1.8 p. 55 #3-5, 8-10, 30.
Section 3.3 p. 157: Syllabus HW
Worksheet #3B (p. 12 of packet): #5.

Wednesday Feb 1: Questions on chain rule. Role of first and second derivatives: second derivative and concavity, and whether tangent line approximation is larger (f CD) or smaller (f CU) than the actual value.
Begin product, quotient rules (Section 3.4).

HW for Thursday, February 2:
WS #5, #1-3 (or review Exponential growth Section 1.7).
WS #3B #4 (product rule)
Section 3.4 #1-17 odd (product rule).

Thursday Feb 2: Questions, Quiz 2 (see below)

HW for Monday, February 6:
Read Section 3.4.
Do syllabus HW p. 161: odd #1-31, even #1-18 (you need to know how to do these, doing all will give good practice, but you don't need to do each one if you understand this topic well).
p. 161 #42, 44-47,49-52,53-56.
WS #3B p. 11 of class pack: #5,6,7. (some of these, especially #6, are more interesting than the problems in Section 3.4), as they combine methods.

Week of February 6 Product, quotient rule (questioins). Periodic (trig) functions and their derivatives: Section 1.10, 3.5. Local max and minima, inflection points (Section 4.1,4.2), Global Max-Min (Section 4.3).

Monday February 6 Quiz 2 back, went over Quiz 2. Product, quotient rule (again, also combined methods).
Periodic (trig) functions overview (Section 1.10). Amplitude, period, vertical displacement, basic trig functions (sine, cosine, tangent) of an angle in terms of adjacent side, opposite side, and hypotenuse of right triangle.

HW for Wednesday Feb. 8:
See HW on product rule for Monday Feb 6.
WS #3B #6,7.
Text: Read Section 1.10. p. 68 #1-5,8,12,13,17.

Wednesday February 8 Questions on Perodic functions, WS #3B #6C,7. Begin local max min, using calculus in graphing, second derivative test. (Section 4.1,4.2).

HW for Thursday Feb. 9 (first part)
Text: Read Section 3.5. p. 165 Odd #1-19, 23,26

HW for Thursday Feb. 9 (second part)
Read Text Section 4.1. Do syllabus HW.

Thursday Feb. 9: Local max-min, continued. Inflection points, graphing curves using calculus.

Homework for Monday February 13:
Section 4.2 p. 186 syllabus HW: Note especially graphing problems #11-15.
Prepare Exam 1A, p. 45-46 of Class Packet as review for Exam 1. Answers later in packet. (Also expect exponential growth problems as on Quiz #2.)

Week of February 13 Review of graphing using calculus, global max-min Section 4.3, Review for Exam 1 (part of Monday, all Wednesday Feb 15 class).

Monday February 13 Graphing using calculus. Global max-min on an interval (section 4.3: making a table of values including critical points and endpoints of the interval, to decide the global max-min. Also #15 p. 191.

HW for Wednesday Feb 15: Section 4.3 p. 191 #1-3, 5-9,15,17-19.
Prepare for Exam 1 (see Exam 1a in packet)

Note: Office Hours and Tutoring My next office hours are Wednesday 1:45-4 PM. However, the Math Workshop in 540 B Nightingale is open and generally not too busy on Tuesday to just drop in. In the Tues AM a math grad student (Marcus), and in PM 1-4 a math grad student who has taught Math U141 (Beth) are tutoring.
The Math Workshop is open late on Wednesday (see your packet for hours). In general one signs up for tutoring on a weekly list (up to 2 hrs/wk).

Wednesday Feb 15 Questions, solve problems from Fall 2005 Exam 1 (not in packet).

Thursday Feb. 16. Exam 1.

HW for Wednesday Feb. 23
Section 4.3 p. 191 #28-30, 36-38,40,43-46.
(optional): Read Section 4.7 in preparation for class.

Week of February 21 Some applications of global max/min. Logistic equation, Surge function and drug concentration (Sections 4.7,4.8). Max-min word problems (Class packet).

Monday February 21 President's Holiday: no class.

Wednesday Februray 22 Exam 1 back, solutiions. Went over exam. Max-min problems from Section 4.3 #36-38,#40 (surge function Section 4.8). Begin max-min word problems (Packet WS #4C, p. 15).

HW for Thursday February 23:
Finish HW in Section 4.3 listed above for Wednesday Feb 23.
Read Worksheet 4C, try #2 at bottom of page.
If time, read WS #4D,E.

Thursday February 23: Max-min word problems, continued (packet). We completed Worksheet 4C (p. 15) #1-3, and p. 21 of packet (MM#1) #3,8.
Passed out information sheet about partial make up (see below: coming events).

HW for Monday, February 27.
Class packet p. 21 (MM#!): #1-6, 7*,8-10.
Section 4.3 p. 192 of text: #31-33.
Read WS #4D,E, and p. 19-20 of packet.
Read Section 4.8, do p. 225 #1,2,6.

Week of February 27 Continue max-min. Surge function (Sect 4.8). Begin integral (Chapter 5).

Monday February 27 Max-min problems involving tickets, yield from orchard. Surge function. Integral as signed area.

HW for Wednesday March 1:
p. 22 Class Packet: (MM#2), #19-22,24,25,29.
Read Section 4.8, p. 225 #1-4,6,10.
If time, Read Section 5.1 (Prepare for Wednesday class).

Wednesday, March 1 Questions on max-min packet p. MM#2, #19,21,22.
Integral. Finding integrals geometrically (as signed area). Worksheet #6A, #1,2. Left and right sums ,trapezoid estimate (begin).

HW for Thursday March 2:
Worksheet #6A, problem #3.
Read Section 5.1. Section 5.1 p. 240 #1-5.

Thursday, March 2 WS MM#2 #23. Then Integral, continued, using Section 5.1 and related exercises p. 240 #4-7. Meaning of integral as signed area, integral of velocity as distance travelled. Left, right, and trapezoid sums (section 5.2).

Homework for Monday March 13.
Text Section 5.1 p. 240 #11, 15,16,19.
Text Read Section 5.2: p. 247 #2.3,5,8,16.
Section 5.3: p. 253 #3-8,11 (some added on web).
To prepare for Quiz 3: See Quiz 3.5 p. 41 of Class Packet, but note we have not discussed the antiderivative yet (so omit #3b).

Have an enjoyable and safe Spring Break!

Week of March 13 Integral, continued; interpretation of integral, fundamental Theorem (Section 5.4). Begin antiderivative (Section 7.1).
Quiz 3 on Thursday.

Monday March 13 Left, right Riemann sums, trapezoid sum (section 5.2). Interpretation of integral of rate as change in amount (Section 5.4).

Homework for Wednesday March 15
Section 5.4 p. 258, #1,2,3-7, #8 and #9 (set up only), 10, 12 (set up only), 14,16,24,29,30-32.
Section 5.2 p. 247 #1,5,6,14,16,17. (added on web).

Wednesday March 15 Questions. Fundamental Theorem and rules for integral (Section 5.5, also p. 272). Antiderivative (begin Section 7.1, if there is time).

Thursday March 16 Group half quiz 3, whole class time. New groups assigned to begin Monday.

HW for Monday
Section 5.5 p. 264 #11-13 (fundamental theorem)
Read Section 7.1 (antiderivative). Do p. 303 #1-7,15,16,63, using the formula for antiderivative of a power: antiderivative of x^n is x^{n+1}/(n+1) +C.

Week of March 20: Area between curves (section 5.3) (waited till having the fundamental theorem). Formulas for integral (p. 272).
Reasoning for fundamental theorem: Uses integral as area and the definition of derivative. (p. 271-272).
Antiderivatives (Section 7.1). Integral and flow (Worksheets 7A,7B).
Finding definite integrals (Section 7.3).
Motion problems.

Monday March 20 HQuiz 3 back, went over #4,5. Second groups.
Area between curves: integral of big-small (section 5.3 p. 252).
Properties of integral: p. 272.
Antiderivatives (Section 7.1): power functions, exponential, ln x, sin (ax), cos(ax).

HW for Wednesday March 22
Section 5.3 p. 254 #25,26,27,30 (note #26,27 added on web).
Class pack p. 41 #2, p. 42 #2.
Section 7.1 p. 303 Odd #1-31.

Wednesday March 22 Definite integral and fundamental Theorem (Section 7.3).
Motion problems (WS #4A exercise 1, WS 4B #1). Relation among position s, velocity v, and acceleration a (differentiate to go down s to v to a, antiderivative to go from a to v to s).

HW to begin for Thursday March 23:
Section 7.3 #1-12,26-33,44-45.
WS #4A Class Packet p. 12 Ex. #1, begin Ex #2.
WS #4B p. 13 #1,2.
p. A2 (Class packet p. 25) #7,10 (accel from gravity on earth is -32 ft/sec^2).

Thursday March 23. Motion, continued. Antiderivative problems (WS 4A #2, WS 4B #2B), Packet p. 26 #11. Interpretation of integral, continued.

HW for Monday March 27 Text Section 7.4 p. 317 #1,2,3,5,13,15-16.
Classpack p. 25-26 (AA) +#1-8,10,11-18.
Classpack p. 40 Quiz 3 #3, p. 47 Exam 2 #2.
Classpack p. 13 WS 4A #2,3.

Week of March 27 Differential equations (chap. 10).

Monday March 27 Check HW. Begin Differential Equations, Exponential growth (Sections 10.1, 10.4).

HW for Wednesday March 29 Section 10.1 p. 401 # 4-6
Section 10.2 p. 404 #2-4.

Wednesday March 29 Diff. Equations, continued.

HW for Thursday: Section 10.4 p. 416 #1-3,9,13,16-18.
Note: in each problem you find k using the solution y= y(0)e^{kt}, and substituting the data given you. for example in #14,you would set
y(3)=y(0)e^{k3}=0.2 y(0), so e^{3k)=0.2, k = ln(0.2)/3. This language is distinct from that in Section 10.1, 10.2 where k is given in the differential equation directly.

Thursday March 30. Worksheet/problems. Newton's law of cooling. (Prof. Eigen, Asst Chair of Math Dept)
Note: I am out all day March 30, Prof. Eigen will give the class. In place of office hours, please feel free to use the Math Workshop in Nightingale Hall. I will have normal office hours next week.

HW for Monday April 3 Section 10.5 p. 424 #1-2,5,6,16-18,25-16,28.

Week of April 3 Newton's Law of Cooling. Practice for Exam 2. Exam 2 Thursday April 6 (for further info see below under "coming events").

Monday April 3
Newton's law of cooling applied to drug level problem (Class pack p. 43 #3);
Second fundamental theorem (text p. 271-272, and sample exam 2 problems as classpack p. 48 #7A);
Problems involving flow of suspended substance in liquic (Class pack p. 48 #5B. for further depth see also WS #7A p. 31 of classpack, Examples 3 and 7A.)
Practice for Exam 2.

Wednesday April 5 Questions, practice for Exam 2

Note: corrections to answers p. 87 in Class Pack (to Exam 2, 2003, p. 48): (thanks to Alison Coll and Young-Jee Kim)
#6: the factor (15-2x) should be (15-x).
#7B. There is a missing (0.5) factor in Trap(4). The actual answer is half that given, since 0.5 = (2-0)/4 is the base of each trapezoid. The format given is a short format for the trapezoid sum, it may be simpler for many students to write out each of the 4 terms.

Note There is a solution to the oyster problem p. 64 #8 (differential equation) on p. 100 of the Class Packet.

Thursday April 6 Exam 2 (see below for more information).

Monday April 10 No class, I was in hospital April 8-14, and could not arrange substitute till Wednesday.

Wednesday, Thursday April 12-13 Mr. David Long gave review, using the final exam from 2004 Spring (?).

Wednesday April 19 I went over topics involving memorization of formulas, or preparing for the final exam, that were different --- or variations from --- the sample final exam material in the Class Pack. These included:
a. Using calculator to find definite integrals
b. Solution to Newton's law of cooling. Like #5 on Exam 2, see solution. The formula is y'=k(A-y) has solution y=A+Ce^{-kt}. Here A is the equilibrium value, and C a constant that can be determined from the value y(0) or one other value.
c. Approximation of derivative at a point P using average slope to a nearby point (see last problem on Worksheet #2A).
d. Motion problems where the velocity is not constant: velocity is integral of acceleration (there is a constant of integration), position is integral of velocity. (there is another constant of integration)

OFFICE HOURS IN NEXT DAYS:
Thurs April 20 11:45 - 12:30 PM
Monday April 24: 12-1, 3-3:30 PM

I will be at math dept grading with other instructors Wed April 26 9:30 AM-1:30 PM or so, and Thursday April 27 9:30 AM - about 12:30 PM or so. These are not office hours, but you can find me.

POSTING OF GRADES: I plan to post course grades late Thursday or Friday of next week, April 27 or 28, outside my office, 526 NI.



Coming events :

FINAL EXAM DATE, TIME
The Math U141 Final Exam is scheduled for Tuesday April 25 at 3:30 PM in 135 Shillman. Bring Extra Credit HW if you wish it to be considered (see below).
Don't miss the final exam, but if you do, you need to contact me as soon as possible, as the course grade without final exam is F.
Of course, a medical or family emergency would be considered.


Announcements:

EXTRA CREDIT HOMEWORK: If you wish your homework to be considered for the extra credit HW grade (helps the quiz grade), bring your homework, or notebook with homework in, if possible with pages and sections of HW labelled, to the final exam. I will give it a letter grade based on amount of HW done, and please take your notebook/HW home after the final exam.

Archived announcements:

Exam 2 April 6.
Max-min, integral, differential equations.
To prepare for Exam 2, see the old exam 2 p. 47ff in Class Pack (except #7C), also p. 51 Sample Integral Problems, and Q4 p. 42 (except 3B,C), and HQ4 p. 43 (you will go over this thursday March 29 or Monday.). Also see problems from HW involving the integral and graphs of a rate of change, as Section 7.3,7.4.

Quiz 3: Thursday March 16, on work since Exam 1. Section 4.8, Chapter 5. Max-Min problems (see Quiz 3 p. 40 #1, and MM #1 HW), Integral (see Quiz 3.5 p. 41 of packet, and Quiz 4 #1). Left, right, trapezoid sums, interpretation of integral.

Partial Make up for Exam 1: Wednesday March 1, 1-3 PM, and/or Thursday March 2 3-4 PM. Meet in my office 526 NI. (we will clarify the times in class). You may attempt 1.5(98-score) problems that you got wrong, on a new exam. You will receive (2/3) (new points) added to your score. MAX new score is 98, a B+ (89% after scaling).
For example, if you received an 80, then you may attempt 27 points you missed (you may have to do more, if these are part of problems), and will receive 2/3( new points), so a maximum of 98 (B+).
Entrance to make up: do corrections to your Exam 1, you must bring old exam 1 and choose with me which problems you wish to redo on the new exam. If you do this before the actual make up day you will save time and help eliminate the one bottleneck.

Exam 1: Feb 16. See Exam 1A, p. 45-46 of Class Packet (answers p. 81ff oF packet). Also expect exponential growth problems, linear approximation along the tangent line as were on Quiz #2.

Quiz 2: Thursday February 2. For sample go over HW since Quiz 1, as well as problems #2-end of Quiz 2 p. 37-38 of packet (answers later p. 75-77in packet), and also Worksheet #5 p. 27 of packet (problems involving exponential growth and composite functions. For these see also Section 1.7 of text).
Class requested an extra credit question on product rule. Pollywogs will appear.

Quiz #1: 40 minutes, Thursday January 19: See Quiz #1B passed out in class, and also Quiz #1 and solutions in Class Pack. Also expect questions related to the HW in text for Wed Jan 18 (see above). For example, graphing the slope function y'=f'(x) given y=f(x). See Section 2.2 of text, also Quiz 2A p. 37 of Classpak, #2B.



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

Math U141 Spring 2006 Information (html) Information, Office hours for Prof. Iarrobino sections

Math U141 home page (html)

Math U141 Fall 2005 Classnotes (may give an idea of timing of future material, quizzes, exams).

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 9, 2006. Last modified: April 19, 2006. URL:http://www.math.neu.edu/~iarrobino/AIMathU141.Spr06.classnotes