Elementary Calculus I

Mathematics 69.107, Sections K and L ... Winter, 1997

HTML 3.2 checked!HTML 3.2 Checked!


Week of Jan. 6-10 Weekly Quiz 1 No Tutorials No assignments
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 Mar 3-7 Weekly Quiz 8 Tutorial 5 Assignment 3
Week of Mar 10-14 Weekly Quiz 9

Review Questions

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
Week of Mar 31-Apr 4 Sample Final Exam Tutorial 8 No Assignment

Detailed class schedule

Dr. Angelo B. Mingarelli,
Herzberg Physics Office #4250
Tel/Fax: (613) 520 3534
Electronic mail

Calculus-Single Variable-Early Transcendentals, by J. Stewart (packaged with Chapter 15 and Student Guide), Third Edition (Brookes/Cole 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.

Third Edition: Sections 2.1-2.6 & 2.10, 3.1-3.6 & 3.8, 4.5 ,5.1-5.5, 6.1 & 6.2, 7.1, 7.2, 7.4, 7.8, 8.1

The prerequisites for this course are:
(1) Ontario Grade 12 Advanced Level Mathematics, or Carleton University 69.006*, or former Ontario Grade 13 Functions,
(2) Ontario OAC Calculus, or Carleton University 69.007*, or approved equivalent. Students who have not passed the prerequisite course may be automatically de-registered during the term. Those that have done poorly in the prerequisites are strongly urged to take 69.007* before attempting 69.107*. Do get advice from the instructor or from the Mathematics Undergraduate Advisor Ken Small, in 4380 Herzberg Building.

Your grade will be calculated either as:
(i) Term Mark 40%;
(ii) Final Examination 60%
(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.

The term mark will be derived from:
(a) 8 or 9 tutorial problem sets (10/40): best 5 of 9;
(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.

The supplemental examination for this course is the final examination for 69.107* in 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.

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

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.


Monday, Jan. 6, 1997


Mondays, 360 Tory Building, 11:30 a.m.
Tuesdays, 360 Tory Building, 1:30 p.m.
Thursdays, 360 Tory Building, 12:30 p.m.


All tutorials are held on Tuesdays, at 12:30 p.m. and will begin on Jan. 14 in various locations depending on the first letter of your family name (surname, cognome(n)):

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

WINTER BREAK: Feb. 24-28, 1997

CLASSES END: Monday, April 7, 1997


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:

Detailed Class Outline

Winter 1997

1 Jan. 6-10 None 2.1, 2.2, 2.6 What is Calculus? Derivatives: Definition, rules, implicit differentiation
2 Jan. 13-17 Assign.1 given 2.4, 2.10, 3.9 Derivatives of trigonometric functions, Newton's method, L'Hopital's rule
3 Jan. 20-24 Assign.1 due, 3.1, 3.2, 3.3,
3.4, 3.5, 4.1-4.5
Exponential and logarithmic functions,
curve sketching
4 Jan. 27-31 TEST 1
Tutorial room
3.5, 3.6, 3.9 Logarithmic differentiation
using L'Hopital's rule,
Exponential growth and decay,
simple differential equations
5 Feb. 3-7 Assign.2 given 3.6 Inverse trigonometric functions, word problems
6 Feb. 10-14 Assign 2, due 4.8, 5.3-5.5, 7.8 Definite Integrals, Fundamental theorem
of Calculus, numerical integration
7 Feb. 17-21 Test 2
Tutorial Room
7.3, 7.5-6 Integration by substitution
(change of variable)
Feb. 24-28
8 Mar. 3-7 Assign. 3 given 6.1, 7.2-3 Trigonometric integrals
Area between curves
9 Mar. 10-14 Assign. 3 due 6.2-3 Integration by parts
Volume by cylindrical shells
10 Mar. 17-21 TEST 3
Tutorial Room
6.2-3, 7.2 More volumes, trigonometric integrals
11 Mar. 24-28 Group Tutorials 7.4, 15.1-2 Partial fractions, separable differential equations
12 Mar.31-Apr.4 Group Tutorials 15.3-4 More differential equations
R E V I E W or enrich

Notes for the Week of Jan.6

Things to remember: 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.

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

Notes for the Week of Jan.13

Remember that you should be careful in deciding what initial point xo to use in Newton's method. If the point is too close to a critical point of the given function, the iterates may not converge to the required root or they may not converge at all!

Also remember that if we are looking for roots of the equation f(x) = c, where c is some constant, then we should be looking for roots of the new equation g(x) = 0 where g(x) = f(x) - c, before we apply Newton's method to g(and NOT f!).

Remember not to apply the Rule of Bernoulli-L'Hospital to quotients of determinate forms ... this may give incorrect results as we saw in class.

Unfortunately, it is not possible for us to fill in the gaps in pre-calculus math. 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.

Notes for the Week of Jan. 20

The whole notion of an inverse function is central to the study of Calculus. As you have gathered already, Euler's exponential function and its, inverse, the natural logarithm have very useful properties, properties which will resurface later in our study of first order differential equations.

Much modeling in the physical sciences involves these two functions; in class we saw their application to the extinction of species (the Seaside Dusky Sparrow) and similar calculations apply to more general populations. For more Calculus resources on the Web check out the site Calculus Resources Online.

Notes for the Week of Jan. 27

There are tons of applications of the exponential functions to real-life problems; e.g., from radiocarbon dating to population growth, and to drinking coffee and interest rates. Calculus is everywhere around you, embedded within the framework of nature. Totum hoc philosophari.

Notes for the Week of Feb. 3

Most assignments and tutorials on this page have answers insertedinto them. Please refer to these for reference and study purposes. Don't forget the test in two weeks.

Notes for the Week of Feb. 10

Test next week: Study online sample quiz and other materials on this page. Items to be studied include exponential growth and decay, simple differential equations, inverse trig. functions, word problems, definite integrals and the Fundamental Theorem of Calculus.

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 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!

Notes for the week of Feb. 17

The Fundamental Theorem of Calculus is by far one of the most useful tools you'll ever need in the applications of Calculus. One of the finest results obtainable is one dubbed "Leibnitz's Rule" which is useful in differentiating an integral with variable limits. For example, in order to find the derivative of the function H(x) defined by the integral of f(t) over the interval a(x) and b(x)...(assume a, b are differentiable functions, etc.) let us write this as:

H(x) = Int(f(t)dt, u(x), v(x))

... we obtain after repeated applications of the Chain Rule and the example we did in class, that,

H'(x) = f(v(x))v'(x) - f(u(x))u'(x)

This includes the Fundamental Theorem of Calculus too, right? Click here for a great link to Leibnitz's Rule...You can practice the Rule there in its most abstract setting. Don't know how to "anti-differentiate" a function?? No problem, just try out this link and the server will do it for you! More topics in first year calculus can be found at the homepage of the MathServ Calculus Toolkit

Notes for the week of Feb. 24 Have a restful term break.

Notes for the week of Mar .3 Having trouble integrating some functions? In order to check your answersand the book's answers for that matter, check out the website of the Integrator (sic). It uses the software Mathematica in order to evaluate the integrals in closed form.