Martin Flashman's courses- Math 120 Spring,'06

Martin Flashman's Courses Math 120 Calculus II Spring, '06

MWF 09:30am-10:25am Fowler 309
and Thursday May 4!
Optional review Session Friday TBA

Final Exams:
Checklist

5-8 Final Exam 1:00- 4:00 pm
in class room

5-9 Alternate Final Exam 1:00-4:00
in Lab room

Lab 1 01:30pm-02:55pm T FOWLER 307
Lab 2 03:00pm-04:25pm T FOWLER 307

$$ Course Description
$$ Course Assignments: Text Problem lists (most recent)
Winplot Materials: Winplot (freeware for PC's that we will use) may be downloaded from Rick Parris's website or directly from Winplot .

Want to find out what your learning style is? Here is an interesting learning styles inventory on the web: NC State
Some advice on how to suceed in a calculus course (at the UNL)
Success in Mathematics (St.Louis University)
Actions of a Good Student (Advice from Prof. Biles - HSU)

WWW Interactive Mathematics Server- on-line and popup calculators for calculus!
Calculus web sites for surfing.
Calculus on the Web [Excellent!] (Temple U)
Visual Calculus (Univ. of Tenn.) calculus (first year) tutorials, etc.
Calculus resource page from Stewart's Calculus Textbook.
Online Mathematics Tutorials at Harvey Mudd College! many excellent tutorials...Calc I-III, DE's, Linear Algebra.
Practice problems from Calculus@internet
Applied Calculus (many good summaries, examples, and tutorials)
Mathlets: Excellent Java applets for Calculus I-III, etc.
SOS Mathematics - Calculus materials
Math 1002 calculus at University Of Aberdeen: Calculus Notes from Ian Craw.

Back to Martin Flashman's Home Page :)
Last updated: 1/22/06

Tentative Schedule for Math 120- Class and Labs - Subject to Revision.
Week	Monday	Tuesday (Lab)	Wednesday	Friday
1. Review Indefinite Integrals	1-23 Introduction and initial review for derviatives.	1-24Review- DE's and estimation, Direction Fields/ Euler's Method/Integral Curves Anti-derivatives, IVP's Linear Estimation Winplot - and DE's	1-25 Indefinite Integrals Core Functions and Linearity	1-27 Substitution
2. The Definite Integral	1-30 More substitution, DE's and Euler.	1-31 Estimating Areas	2-1Euler meets area-The Fundamental Theorem of Calculus I	2-3 The Definite Integral- Definition - Interpretations The Fundamental Theorem of Calculus I'
3.More about the Definite Integral	2-6 Average values	2-7 Areas and definite integrals.	2-8 Areas between curves.	2-10 Work
4. Misc. on Def. Int.	2-13 More work. Start Volume	2-14 More Volume	2-15 Volumes	2-17 Substitution w/ Def. Int. Begin Numerical Integration
5. Numerical Integ.	2-20 Presidents Day- no class	2-2 Numerical Integration	2-22 Simpson's Rule	2-24 Arc Length SC:VIII.B
6. More Applications	2-27 Arc Length VIII.B General planar curves.	2-28 Exam 1	3-1Using Functions to estimate Integrals.	3-3 A preview of Taylor theory. Sensible Calculus IX.A (On-Line)
7. DE's and integration	3-6 Properties of the Definite Integral	3-7 Models and de's continued Taylor's Theorem I (exp)	3-8 Taylor's Theorem I (exp) and estimating int_0^1 exp(-x^2) dx	3-10Models and DE's define functions.FTof C for DE's. Learning and other rates that decrease over time: arctan.
	3-13 Spring Break- No Classes	3-14 Spring Break- No Classes	3-15 Spring Break- No Classes	3-17 Spring Break- No Classes
9. DE's and Taylor	3-20 FTof Calc (DE's) Separable DE's	3-21 Integrals Definite and Indefinite- How they fit into solving DE's	3-22 Sensible Calculus IX.B (On-Line) Taylor's Theorem II	3-24 TT II
10.Series Testing	3-27 IX.B (On-Line) Calculus for TT IX.C (On-Line).	3-28 Taylor's Theorem	3-29 The Logistic Taylor III Introduction to sequences, and convergence.	3-31
11. Power series	4-3 Geometric Series	4-4 Sequences and Series- spreadsheets and graphs	4-5 Harmonic and Power Series	4-7Integral and Comparison testing Alternating series.
12. Taylor Series Plus	4-10 Positive Series. Integral test begun	4-11 Midterm Exam #2	4-12 Infinite integrals, Integral test again. Integration by parts	4-14 Integration by PartsII, Power Series. Taylor Revisited and reviewed.
13..Integration Methods	4-17 Abs . converg. & Ratio test Power series II Applications of Power series to DE's.	4-18 Power Series and De's	4-19 Finish Ratio Test and Applications to Taylor Series and DE's. Power Series Theorem -Differentiation/ Integration.	4-21 Last Breath on Series?Examples...Proofs?
14	4-24 Arc Tangent	4-25 Integration Gateway Test.	4-26 Improper Integrals II	4-28 More Arctan stuff L'Hospital's Rule
15	5-1 Volume of a torus Application of Improper Integrals Area and The Normal Curve(review of integration!)	5-2 Overview of Course! Open Problem Session.	5-3 Simple algebra for series. Fourier Series? Misc. Methods (partial fractions)? ln(2)- Newton's computation/series? Calculus and proabability- darts?	5-4 Thursday! Last class Some last observations on the semester's work. What about the final exam!
16 Final Exams	5-8 Final Exam 1:00- 4:00 pm in class room	5-9 Alternate Final Exam 1:00-4:00 in Lab room

Spring, 2006 MATH 120 : CALCULUS II M.FLASHMAN
Tentative Assignments-This will be revised further! [1-22-06]
(Text: SM = Smith and Minton, 2nd Ed. / SC = Sensible Calculus online materials) and recommended problems(tentative- subject to change!)

Date Due

Reading

Problems

Optional

Viewing: Ed Berger CD Tutorial [# of minutes]
* means optional #means on-line report on Blackboard

HW #1
1-27

SC IVD
SC IVE (on-line)
S&M:6.6
Math 110 Final Solutions

Ch Reviews
p238:1,2,23,31,37, 75
p319:1,49
553: 71, 75 a

A tutorial on slope fields with an interactive JAVA applet to explore slope fields.

Calculus I in 20 minutes - watch only the first 15 minutes! Last five are a preview for the next 2 weeks! 9.1.1.
Antidifferentiation [14]
#9.1.2. Antiderivatives of Powers of x [18]
#9.1.3. Antiderivatives of Trigonometric and Exponential Functions [10]

HW #2
1-30

SC IVA(On-line)
SC IVB (On-line)
S&M: 4.1 pp322-328

On line IVA:1(a,d,e,f),10; 4, 5(a,b),8,11
S&M: 4.1 p322: 5-11; 15-21odd;55-57;67, 68

p334:79
A java graph showing
f (x)=P'(x) related for f a cubic polynomial
A tutorial on antiderivatives and indefinite integrals.

9.2. Integration by Substitution
9.2.1. Undoing the Chain Rule
9.2.2. Integrating Polynomials by Substitution
9.3.1. Integrating Composite Trigonometric Functions by Substitution

HW #3
2-1

S&M: 4.6 pp374-378
SC IV.F READ

S&M: 4.6 p382:5-8,11,13,16,21,26,29,39

On-line tutorial for Substitution
Another Tutorial on substitution.

HW #4
2-3

SC IVF(On line)

IV.F: 1,3,5,13,15,17(on-line)

9.4.1 Approximating Areas of plane regions.

HW #5
2-6

S&M: 4.2 p334- Example 2.5
SC VA ( On Line)

S&M: 4.2 p 340: 7-12
V.A: 1,2 a (on line)
S&M p 372: 5-7, 13,15,

A tutorial on summations and summation notation.

HW #6
2-8

S&M: 4.4 pp359-361
5.1 pp402-405

S&M p 373:77-82

18.1.1 Finding the Average Value of a Function [8]

HW #7
2-10

S&M: 5.1 pp402-405
S&M:5.6 pp453-454

S&M p 409: 5,7,9-13

9.4.4 Illustrating the fundamanetal theorem of calculus[13]
9.4.5 Evaluating Definite Integrals [14]
10.2.1 The area between two curves [9]

HW
#8
2-13+15

S&M:
5.1 pp 405- 407
5.6 pp453-454

S&M p 409: 13,17, 27,29
S&M p462 :5, 11 (wait till 2-15)

10.2.2 Limits of integration and area[15]
10.2.3 Common Mistakes to Avoid

Summary #1
2-14

Partnership Summary #1 covering work through February 8th should be submitted by 5 pm
[2pages - 1 side or 1 page -2 sides.]

HW #9
2-(15) 17

S&M
5.6 pp453-454
5.2: 411-418

S&M p462 :5, 11
S&M p423: 5, 19, 20, 21a, 35

18.6 Work ( 3 segments) [4 + 5+5]
18.2 Finding volumes using cross sections [9+12]

HW #10
2-17

S&M
5.2: 411-418

S&M p423: 211, 23a, 31 a,b

HW #11
2-22

S&M
4.6: pp380-381
4.3: p 347-348
4.7: pp 384-388
SC VA ( On Line)

S&M p382:47-51,54
S&M p349: 11-13,35,36,41,42
S&M p396 part a and b only for 9 and 10

HW #12
2-24

S&M
4.7 pp389-392 remk 7.3.
SC V.D (on- line)

S&M p396: 13, 31-33,37

16.9.1 Deriving the Trapezoidal Rule [12.5]
18.5.2 Finding Arc Lengths...[14]

HW#13
2-27

5.4 pp434-435
Probability

S&M p440: 5, 9

SC:Arc Length VIII.B

Exam I
2-28

Midterm Exam I covers material related to HW's 1-12.

Optional Review:
p398:1-11 odd; 21-25odd; 31,41,45,49 odd, 61
p 475 3,7, 11, ,15a, 33
p553: 71,75a.

HW #14
3-3&8

4.4 p351, 356-360
9.3 pp739-742
Sensible Calculus IXA (On-Line)

S&M: p745: 5, 9
p362:31-34
SC IXA: 1,2

HW #15
3-10

4.4 p351, 356-360
Sensible Calculus IXA (On-Line)

SC IXA: 3-5,10[a-e]

HW #16
3-22

4.5 p 367-269
6.4: p503-505
6.5: p 512-516
516-518?

S&M: p372: 41-43, 55, 57
S&M: p 509: 5,6; p 519:13,17-19, 29

SC:VI.D models and inverse trig

HW #17
3-24

6.4ex 4.1
Sensible Calculus IX.B (On-Line)

S&M: p510: 13-15
S&M: p 519: 19,20 29,33
Sensible Calculus IX.B (On-Line) :1-5

20.3.1 Exponential Growth [12]
20.1.2 Solving Separable Differential Equations [9]
20.1.3 Finding a particular solution. [6]

HW #18
3-27

6.4 ex 4.2
IX.B (On-Line)
In advance for class: IX.C (On-Line).

S&M p 510: 25-27, 29
Sensible Calculus IX.B (On-Line) :13-15

20.3.2 Radioactive decay [8]
19.12.2 Maclaurin Polynomials [9]
19.12.1 Taylor polynomials [14].

HW#19
3-29

IX.C (On-Line)
6.4 pp506-8

IX.C (On-Line):1-5,7,8
S&M p 510: 31,33,35

16.6.1 Introduction to Partial Fractions [13]

Summary #2
3-29

Partnership Summary #2 covering work through March 24th should be submitted by 5 pm
[2pages - 1 side or 1 page -2 sides.]

HW #20
3-31

6.5: p516-518
IX.D ;X.A

S&M p519:35, 37
IX.D: 1-4, 10,12

p520:55, 59

HW #21
4-3

X.A
X.B
8.1 pp622-623, 625-626

X.A : 1-9 odd
S&M: p634 5-8, 9-14 part a only, 15-21

19.1.1 The limit of a sequence.[10]
19.1.2 Deteremining the limit of a sequence.[9]

HW #22
4-5

X.B
8.1 pp628-631
8.2 pp 636-641

S&M : p 634:51-53
S&M: p644: 5-10, 19, 20, 25

19.3.1 Introduction to Infinte Series [11]
19.3.3 geometric series[13]

HW #23
4-7

X.B
8.1 pp627-8;631-632.
8.2 pp641-644

S&M : p 634: 39-41;
p644: 13,15,16,25-28

19.4.1 properties of convergence [7]
19.4.2 test for divergence [8]

HW #24
4-10

8.3: pp 647-649
8.4 : pp 658-661

S&M: p656: 5-11odd.;
p664: 5-11 odd, 31

Exam II
4-11

Midterm Exam II covers material related to HW's 13-24.

Optional Review:
p362: 31,33; p400: 59
p476:19; p552: 17, 19, 49, 51, 55, 57, 61, 63, 75
p718: 1-5,9,10,19,21,25, 69

HW #25
4-14

7.2pp560-563
7.7pp610-613 omit example 7.8
8.3 pp647-649 omit example 3.1.

S&M 7.2 p566: 3-11
7.7 p617: 15-17, 25
8.3: p 656: 4-9, 19

VII.C. Integration by Parts

16.6 .1- 3 Integration by parts.
19.5 .1&2 The integral test

HW #26
4-17

7.2 pp564-566
X.B5

S&M 7.2 567: 13,19,21, 25, 31, 32, 41,45

HW# 27
4-21

8.5: p666-670
8.6:675-678
X.B5
XI.A Power Series

S&M 8.5 p673: 5-9, 19, 20, 23, 24
S&M 8.6 p681: 23,24,33,37

19.9.1 Absolute and conditional Convergence [12]
19.10.1 The Ratio Test.
19.10.2 Examples of the Ratio Test
19.14.1- 19.14.3 Power Series
19.15.1 Diff'n and integ'n of power series.

HW #28
4-24

XI.A Power Series
8.7 Example 7.3, 7.6, 7.8
8.8 Example 8.4 and 8.5

S&M: 8.7: p694: 29-32, 39-42, 45-47, 49
S&M: 8.8 p702: 17,19.

HW #29
4-26

6.7 pp530-534
6.8 536-538; 540
p 648 Example 3.1
p679 Example 66.

S&M 6.7 p535: 5-10,13,20, 29
S&M 6.8 p 542: 7, 15; 17, 21, 22,27,28, 33
S&M p 719:75

VI.DMore Models & Inverse Trigonometry

HW #30
4-28

7.7: pp 605-609

S&M: 7.7 p617: 3-9 odd, 11-13,37,38

17.1.Improper integrals

HW #31
5-1

7.6

S&M: 7.6 p603: 3-5, 17,18,23, 25, 26,29

14.1 and 14.2 L'Hopital's Rule

Summary #3
5-1

Partnership Summary #3 covering work through April 28th should be submitted by 5 pm
[2pages - 1 side or 1 page -2 sides.]

Back to Martin Flashman's Home Page :)

OFFICE: Fowler 325 PHONE: (323) 259- 2555
Hours (Tent.): M-F 11:30-12:30 AND BY APPOINTMENT or chance!
E-MAIL: flashman@oxy.edu WWW: http://flashman.neocities.org/
***PREREQUISITES: Calculus I

TEXTS: Required:
Calculus, 2/e Robert T. Smith, Millersville University Roland B. Minton, Roanoke College
Calculus , CD, by Ed Burger- Thinkwell Publishing.(ISBN 1931381992)
Excerpts from Sensible Calculus by M. Flashman as available on the web from Professor Flashman.

Calculus: An Interactive Text

(ISBN 007-239848-5)
This browser-based electronic supplement provides access to the entire Calculus text in an interactive format. Features include over 200 text-specific JAVA applets and over 400 algorithmically-generated practice problems, designed to demonstrate key concepts and examples from the text. The electronic student solutions manual is integrated for full comprehension of exercises.

Note that with the purchase of your textbook, you have access to the Calculus Online Learning Center, which helps students learn calculus with automatically graded practice quizzes, additional explanations of difficult topics and guides for TI calculators. Live tutorial assistance is also available.

Catalog Description:

Calculus at Occidental:
Calculus differs in some respects from the traditional courses offered at some secondary schools and most other colleges or universities. Occidental’s program is based on scientific modeling, makes regular use of computers, and requires interpretation as well as computation. A variety of courses comprise this program, accommodating different levels of preparation. The core content is described below as Calculus 1 and 2. Actual courses suited to different levels of preparation are listed under each description.
CALCULUS 1: SCIENTIFIC MODELING AND DIFFERENTIAL CALCULUS.

Many mathematical models in the natural and social sciences take the form of systems of differential equations. This introduction to the calculus is organized around the construction and analysis of these models, focusing on the mathematical questions they raise. Models are drawn from biology, economics, and physics. The important elementary functions of analysis arise as solutions of these models in special cases.

The mathematical theme of the course is local linearity. Topics include the definition of the derivative, rules for computing derivatives, Euler’s Method, Newton’s Method, Taylor polynomials, error analysis, optimization, and an introduction to the differential calculus of functions of two variables.

CALCULUS 2: SCIENTIFIC MODELING AND INTEGRAL CALCULUS.

This course continues the study of the calculus through scientific modeling. While Calculus 1 is concerned with local changes in a function, Calculus 2 focuses on accumulated changes. Models solved by accumulation functions lead to the definition of the integral and the Fundamental Theorem of Calculus.

Additional topics include numerical and analytic techniques of integration, trigonometric functions and dynamical systems modeling periodic or quasiperiodic phenomena, local approximation of functions by Taylor polynomials and Taylor series, and approximation of periodic functions on an interval by trigonometric polynomials and Fourier series.

Supplementary notes and text will be provided as appropriate on the web.
$$ Try the Backgrounds Check Quiz on Blackboard. If you don't do well on the on-line backgrounds assessment quiz , see me soon!

TESTS AND ASSIGNMENTS: There will be several tests in this course. There will be 2 Gateway Exams, many on-line reality check quizzes, two midterm exams and a comprehensive final examination.

Gateway Exams: You must pass two Gateway exams with a score of 9/10. These exams will test basic skills and concepts needed for the course, and can be taken as many times as needed (subject to target dates below, extension permission required after the target date). One successful Gateway test will count as one on-line quiz.

* Derivatives. February 16th. [Changed -2-3]

We will use Blackboard for on-line reality quizzes. Click here for some information on how to use Blackboard.
You can also go directly to the Oxy Blackboard .
Homework assignments are made regularly. They should be done neatly. Problems from the assignments will be discussed in class.

Exams will be announced at least one week in advance.

THE FINAL EXAMINATION WILL BE SELF SCHEDULED with at least two alternate offerings.
The final exam will be comprehensive, covering the entire semester.
MAKE-UP TESTS WILL NOT BE GIVEN EXCEPT FOR VERY SPECIAL CIRCUMSTANCES!
It is the student's responsibility to request a makeup promptly.

*** DAILY ATTENDANCE SHOULD BE A HABIT! ***

Using the CD Tutorials: Whenever a CD tutorial is assigned, that should be viewed by the due date of the assignment. As part of that assignment, you may be asked to respond to some relevant questions on Blackboard under the CD report title. These questions may be related to the solution of a specific problem, the development of a concept, or the organization of a technique as presented on the CD. Each CD tutorial question will add 5 points to your CD Tutorial point total. CD Tutorial work will be used in determining the 50 course points.

Change: CD Reports were discontinued- the 50 points for this work is reallocated to homework.

Partnership Activities: Every three weeks your partnership will be asked to submit a summary of what we have covered in class. (No more than two sides of a paper.) These may be organized in any way you find useful but should not be a copy of your class notes. I will read and correct these before returning them. Partners will receive corrected photocopies.

Your summaries will be allowed as references at the final examination only.

Every week (with some exceptions) partners will submit a response to the "problem/ lab activity of the week."
All cooperative problem work will be graded 5 well done, 4 for OK, 3 acceptable, or 2 or 1 unacceptable and will be used in determining the 50 points allocated for cooperative assignments.

GRADES: Final grades will be determined taking into consideration the quality of work done in the course as evidenced primarily from the accumulation of points from tests, various individual and "team" assignments.

Midterm exams will be worth 100 points each and the final exam will be worth 200 or 300 points.
Homework performance will count for 100 points. Changed: Now 150 points.
On-line Reality quizzes will be used to determine 150 points.[I will not use the lowest 20% of these scores.]
Cooperative problem assignments (POW's and labs) and summaries will be used to determine 100 points.
The CD tutorial responses will be used to determine 50 points. [Changed: now 0.]
The final examination will be be worth either 200 or 300 points determined by the following rule:

The final grade will use the score that maximizes the average for the term based on all possible points .


Reality Quizzes	150 points
2 Midterm Examinations	200 points
Homework	150 points
CD Tutorials - discontinued	0 points
Cooperative work(Labs/POW's + Summaries)	100 points
Final Examination	200/300 points
Total	800/900 points

MORE THAN 4 ABSENCES MAY LOWER THE FINAL GRADE FOR POOR ATTENDANCE.
In my experience students who are actively engaged in learning and participating regularly in a variety of activities will learn and understand more and retain more of what they learn. Each component of the course allows you a different way to interact with the material.

** See the college course schedule for the dates related to the following :

	Last Day to Drop Courses
	CR/NC Forms Due
	Withdrawal Period Begins
	Last Day to Withdraw From Class

Technology: The computer or a graphing calculator can be used for many problems.

We will use Winplot. Winplot is freeware and may be downloaded from Rick Parris 's website or directly from one of these links for Winplot1 or Winplot2 . This software is small enough to fit on a 3.5" disc and can be used on any Windows PC on campus. You can find introductions to Winplot on the web.

Students wishing help with any graphing calculator should plan to bring their calculator manual with them to class.
Graphing Calculators: Though much of our work this semester will be using the computer, graphing calculators are welcome and highly recommended. The HP48G, HP 49 and the TI-89 and 92 are particularly useful though most graphing calculators will be able to do much of the work. If you would like to purchase one or have one already, let me know. Students wishing help with any graphing calculator should plan to bring their calculator manual with them. I will try to help you with your own technology when possible during office hours or by appointment (not in class).

Use of Office Hours and Optional "5th hour"s: Many students find the second semester calculus difficult because of weakness in their pre-calculus and first semester background skills and concepts. Difficulties that might have been ignored or passed over in previous courses can be a major reason for why things don't make sense now.

You may use my office hours for some additional work on these background areas either as individuals or in small groups. My office time is also available to discuss routine problems from homework after they have been discussed in class and reality check quizzes as well as using technology. Representatives from groups with questions about the Problem of the Week are also welcome.
I will try to organize and support additional time with small (or larger) groups of students for whom some additional work on these background areas may improve their understanding of current coursework.
Regular use of my time outside of class should be especially useful for students having difficulty with the work and wishing to improve through a steady approach to mastering skills and concepts.
Tentatively Karl McMurtry has been assigned to do AMP (Academic Mastery Program) workshops for this class. Karl will sit in on the class, write worksheets on the course material, and hold weekly workshops in which he will help you through the worksheets.

Don't be shy about asking for an appointment outside of the scheduled office hours.

Math 120 Final Topic Check List Core Topics are italicized.

Differential Equations and Integral Calculus
A. Indefinite Integrals (Antiderivatives)
Definitions and basic theorem
Core functions- including Arctangent.
Simple properties [ sums, constants, polynomials]
Substitution
B. Euler's Method, etc.
Euler's Method
Simple differential equations with applications
Tangent (direction) fields/ Integral Curves
C. The Definite Integral
Euler Sums / Definition/ Estimates (endpoints/midpoints) /Simple Properties / Substitution
Interpretations (area / change in position)
THE FUNDAMENTAL THEOREM OF CALCULUS - evaluation form
THE FUNDAMENTAL THEOREM OF CALCULUS - derivative form
D. Applications
Recognizing sums as the definite integral
Areas (between curves).
Volumes (cross sections- discs). Average value. Work.
Differential Equations and Integration
   Tangent Fields and Integral Curves.
   Numerical Approximations.
            Euler's Method.
            Midpoints.
            Trapezoidal Rule.
            Parabolic (Simpson's) Rule.
Integration by Parts.
Separation of Variables.

Improper Integrals: Extending the Concepts of Integration.
Integrals with noncontinuous functions.
Integrals with unbounded intervals.

L'Hopital's Rule: 0/0 inf/inf inf - inf 0*inf 0^ 0 1^inf

Taylor's Theorem.
Taylor Polynomials. Calculus.
Using Taylor Polynomials to Approximate: Error Estimation.
      Derivative form of the remainder.
      Approximating known functions, integrals
      Approximating solutions to diff'l equations using Taylor's theorem.

Sequences and Series: Fundamental Properties.
Sequences.
Simple examples and definitions: visualizing sequences.
         How to find limits.
         Key theory of convergence.
             The algebra of convergence.
             Convergence for monotonic sequences.
Geometric series. Harmonic series. Taylor approximations.
Theory of convergence (series).
     The divergence test.
     Positive series.
          Bounded convergence tests.
           Integral tests.
           Ratio test (Part I).
           Absolute convergence.
             Absolute convergence implies convergence.
     Alternating Series Test.
     Ratio test (Part II).

Power Series: Polynomials and Series.
The radius and interval of convergence.
Functions and power series [derivatives and integrals].

Bonus Essay question for final:

Suppose P(t) is a positive continuous function on [a,b] that gives the velocity at time t of an object moving on a straight line. Explain using the mean value theorem why there is some number c between a and b where P(c) = 1/(b-a) ò_x=a ^x=b P(x) dx.

Interpret this equation with either

(i) a discussion of the velocity and position of the object with the position function given by a definite integral from time x=a to time x=t or

(ii) a discussion of the area under the graph of Y=P(x) above the X-axis from X=a to X=b and the area of a rectangle with height P(c) and width (b-a).

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