# Introductory Calculus

Mathematics 69.007, Section A ... Winter, 1997

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 Week of Jan. 6-10 Weekly Quiz 1 No Tutorials No Assignment Week of Jan. 13-17 Weekly Quiz 2 Tutorial 1 Assignment 1 Week of Jan. 20-25 Weekly Quiz 3 Tutorial 2 Sample Test 1 Week of Jan. 27-31 Weekly Quiz 4 Test 1 No Assignment Week of Feb 3-7 Weekly Quiz 5 Tutorial 3 Assignment 2 Week of Feb 10-14 Weekly Quiz 6 Tutorial 4 Sample Test 2 Week of Feb 17-21 Weekly Quiz 7 Test 2 No Assignment Week of Feb 24-28 Winter Break Winter Break Winter Break Week of Mar 3-7 Weekly Quiz 8 Tutorial 5 Assignment 3 Week of Mar 10-14 Weekly Quiz 9 Tutorial 6 Sample Test 3 Week of Mar 17-21 Weekly Quiz 10 Test 3 No Assignment Week of Mar 24-28 Weekly Quiz 11 Tutorial 7 No Assignment Please ignore error messages Material will be up shortly Week of Mar 31- Apr 4 Sample Final Exam No Tutorial No Assignment

INSTRUCTOR:
Dr. Angelo B. Mingarelli,
Herzberg Physics Office #4250
Tel/Fax: (613) 520 3534
Electronic mail or: angelo@math.carleton.ca

TEXTBOOK:
Calculus: A First Course, by Stewart, Davison and Ferroni et al., 1989 Edition (McGraw-Hill/Ryerson Publishers); available at the Bookstore... and, highly recommended is my...
The ABC's of Calculus by Angelo B. Mingarelli; Module on Inverse Functions, 80 pp., from the Instructor; cost \$12.00 net.

PREREQUISITES:
The prerequisites for this course are:
(1) Ontario Grade 12 Advanced Level Mathematics, or Carleton University 69.006*, or former Ontario Grade 13 Functions.
It is strongly recommended that 69.017 be taken before this course. Do get advice from the instructor or from the Mathematics Undergraduate Advisor Ken Small, in 4380 Herzberg Building.

EVALUATION
(i) Term Mark 40%;
(ii) Final Examination 60%
OR,
(iii)Final Examination 100% whichever is better.
In any event, your final course grade is the larger of the two numbers: A and B where A=(i)+(ii) and B=(iii).
A final examination grade of < 40% results in automatic failure (FNS grade) in this course, regardless of term work.

TERM MARK:
The term mark will be derived from:
(a) 7 tutorial problem sets (10/40): best 5 of 7;
(b) 3 Assignments (15/40):
(c) 3 tests (15/40):

Note: The "best x of y" rules allow you to miss some of the term events for any reason (medical or otherwise).
Only under highly exceptional circumstances will a test/assignment/problem set be postponed to a later date.

SUPPLEMENTAL EXAMINATION
The supplemental examination for this course is the final examination for 69.007* during the summer of 1997. A supplemental examination replaces the final examination mark in the grade calculation. A supplemental examination will not be allowed in cases wherethe term work is unsatisfactory or the final examination mark is extremely low.

CALCULATORS
You may use any non-programmable calculator for the examinations and tests in this course.

WITHDRAWAL
The last date for withdrawal from the course is Mar. 14. If you decide to leave the course before the end of term, it is much better, in terms of your academic career, to formally withdraw from the course than to simply ignore it and get an FNS.

CLASSES BEGIN:

Monday, Jan. 6, 1997

LECTURE SCHEDULE:

Mondays, 3160 Herzberg, 12:30 p.m.
Wednesdays, 3160 Herzberg, 11:30 a.m.
Thursdays, 3160 Herzberg, 1:30 p.m.

All tutorials are held on Thursdays, at 11:30 a.m. and will begin on Jan. 14 in various locations depending on the first letter of your family name (surname, cognome(n)):

• Family names beginning with the letters A - Le; Please go to Tutorial 2: Ms. Xiao, in 505 Southam Hall.
• Family names beginning with the letters Li - Z; Please go to Tutorial 3: Ms. Mallick, in 2115 Herzberg Building.

STATUTORY HOLIDAY: Friday, March 28 ...University closed

WINTER BREAK: Feb. 24-28, 1997

CLASSES END: Monday, April 7, 1997

TUTORIAL CENTRE:

Please note that the mathematics TUTORIAL CENTRE, in Herzberg Physics Building, Room 4385, will be opening on
Monday, Jan. 20
Hours for the center are as follows:
MONDAYS TO THURSDAYS: 10 AM TO 4 PM
FRIDAYS: CLOSED

## Detailed Class Outline

#### Winter 1997

WEEK DATES TESTS SECTIONS TOPICS
1 Jan.6-Jan.10 None 6.1 - 6.5 Review of trigonometric definitions, formulae and identities. Emphasis on use rather than theory
2 Jan.13-Jan.17 Ass.#1
given
1.1-1.5, 7.1 Limits, continuity and trigonometric limits.
3 Jan.20 - Jan.24 Ass.#1
due
2.1-2.4 Definition of derivative, power, sum, product rules.
4 Jan.27 - Jan.31 TEST 1 2.5,2.6,7.2,7.3 Quotient, chain rules, derivatives of trigonometric functions.
5 Febr.3-Febr.7 Ass.#2
given
2.7,2.8,3.1-3.4 Implicit differentiation, higher order derivatives, rates of change.
6 Febr.10 - Febr.14 Ass#2
due
3.5 Related rates. Catch-up.
7 Febr.17-Febr.21 TEST 2 4.1 - 4.3 Increasing/decreasing functions, extreme values and first derivative test.
~ Febr.24-Febr.28 ~ WINTER BREAK ~
8 Mar.3 - Mar.7 Ass.#3
given
4.4,4.5,7.4 Applied extreme value problems.
9 Mar.10-Mar.14 Ass.#3
due
5.1,5.2 Asymptotes, catch-up.
10 Mar.17-Mar.21 TEST 3 5.3-5.5 Concavity, Second derivative test, curve sketching.
11 Mar.24 - Mar.28 Group
Tutorials
8.1-8.4 Logarithm and exponential, and their derivatives.
12 Mar.31-April 4 Group
Tutorials
~ Catch-up and Sample Final Exams
13 April 7-April 11 Group
Tutorials
~ R E V I E W

Things to remember:

There are Netscape browsers available in the Herzberg Building...Click here for further details.

Not every continuous function is differentiable...remember K. Weierstrass gave an example of a function (over 100 years ago) which is continuous at every point of the real line but does not have a derivative anywhere!. If you want to "see" this example, look at the book by E.C. Titchmarsh entitled Theory of Functions, Oxford University Press (1930's).

Remember that fractals also give rise to examples of nowhere differentiable curves. Check out the following URL for a (math. intensive) fractal page. Other sites which exhibit fractals include:

Any Calculus book can be used in conjunction with this course...We only use Stewart for reference purposes, basically.

OK, I am now aware that some of you are having difficulty with basic trigonometry and functions. Unfortunately, it is not possible for us to fill in the gaps in this course and therefore you will have to review your trig. and basic function theory on your own. A fair site on the Internet where this can be done can be found at the University of Saskatchewan's Math. Readiness Site where all sorts of deficiencies in pre-calculus math. can be tested. Perhaps all the sections should be tried there beginning with the Introductory level exercises and proceeding to the more advanced ones. Do try out this week's quiz .

One technique for solving these types of convergence questions about sequences is as follows:

Let's say you want to show that xn -> L, where you've guessed the value of the limit "L".

1. First, you set yn = xn - L.
Now you need to show that yn -> 0, right?
2. Give yourself a number "epsilon" which we call "e" for simplicity. It is small and positive...
3. Write down the inequality |yn| < e
4. Solve the above inequality for n, if you can.
You should get a lower bound for n of the form " n > (something involving "e"), which we call N.
5. This lower bound, N, you get above should depend on the variable "e".
6. Since "e" can be any given number, we can start the whole procedure again for a different "e", and then get a different N, and so on ....

This will prove that the sequence yn -> 0 or (by definition) xn -> L.

About Tutorial 1:All the questions were out of the textbook assigned to this course and answers will appear eventually on this site. Try the weekly quizzes, above, and work them out completely.

Test next week! Review all internet quizzes, weekly quizzes, sample test, and tutorials. Assignment 1 Solutions are posted outside the Math. Tutorial Centre, in HP.4385.

Test this week! See items of the preceding week, in particular the online Sample Test 1 for you to try out.

Assignment this week! Remember that identical assignments will be given the grade of "0": Collaboration is OK but not identical submissions. Work out as many examples as you can from the sections on derivatives, namely, sections 2.1-2.5 covering the Product rule, etc.

Remember that identical assignments will be given the grade of "0": Collaboration is OK but not identical submissions. Try out the sample test 2 in preparation for next week's test.

Nothing much this week but are you interested in graphing software using Java? If your browser is java-enabled Check out this site and you'll see a plotter in action as well as a method for finding the roots of the function you insert (Newton's method? A technique you'll learn about next year). A minor but annoying bug makes the cursor disappear once you click in a window. You may need a few coffees to remember where you were ... just keep counting! We'll have to know this later on so why not let a computer graph your functions for you? (just so you can check your own graphs, of course).

Do have a happy and safe Term Break!

As we're about to embark upon graphing problems don't forget to try the java-powered site mentioned in the Feb 17 Notes, above.