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

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- World Wide Web Browsers on campus
- FAQ about Tutorials in this course
- Notes for the week, Jan 6-10, 1997
- Notes for the week, Jan 13-17, 1997
- Notes for the week, Jan 20-25, 1997
- Notes for the week, Jan 27-31, 1997
- Notes for the week, Feb 3-7, 1997
- Notes for the week, Feb 10-14, 1997
- Notes for the week, Feb 17-21, 1997
- Notes for the week, Feb 24-28, 1997 TERM BREAK!
- Notes for the week, Mar 3-7, 1997
- Notes for the week, Mar 10-14, 1997
- Notes for the week, Mar 17-21, 1997
- Notes for the week, Mar 24-28, 1997
- Notes for the week, Mar 31-Apr 4, 1997

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 Feb 24-28 |
TERM BREAK | TERM BREAK | TERM 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 |

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

**INSTRUCTOR:**

Dr. Angelo B. Mingarelli,

Herzberg Physics Office #4250

Tel/Fax: (613) 520 3534

Electronic mail

**TEXTBOOK**:

*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.

**TOPICS COVERED:**

*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

**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,

(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.

**EVALUATION**

Your grade will be calculated either as:

(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) 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.**

**SUPPLEMENTAL EXAMINATION**

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.

**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, 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)):

- Family names beginning with the letters
**A - Gr**; Please go to Tutorial 2: Ms. Hay, in 311 Steacie Bldg. - Family names beginning with the letters
**Gs - Mc**; Please go to Tutorial 3: Ms. Mallick, in 312 Steacie Bldg. - Family names beginning with the letters
**Md - Sim**; Please go to Tutorial 4: ???????????, in 518 Southam Hall - Family names beginning with the letters
**Sin - Z**; Please go to Tutorial 5: Mr. Dubé in 505 Southam Hall

**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

WEEK |
DATES |
TESTS |
SECTIONS |
TOPICS |

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 2Tutorial Room |
7.3, 7.5-6 | Integration by substitution (change of variable) |

WINTER BREAK |
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 3Tutorial 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 equationsR E V I E W or enrich |

**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.

Remember that you should be careful in deciding what initial point x_{o} 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.

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.

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*.

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.

**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!

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.