Calculus,
2006-2007
Christ
Preparatory Academy
Mr. Rives
Phone: 816-223-5565
Email: steve.rives@gmail.com
Web: www.Math.MrRives.com
Course Description:
This class is for those who fear math – my goal is to demonstrate how winsome and delightful a subject it really is. It is also for those who love mathematics and philosophy, who wish to know more of man’s ability to create systems of thought.
This course is a combination of applied math and mathematical philosophy. It traces the development of both from ancient to modern times. There is a heavy concentration on understanding the meaning of the derivative, limit and integral (all being easy subjects once explained in English). This class is designed to take a student from algebra as a tool, to math as a philosophical system of thought. This includes the study of the infinite and continuity, rational and irrational numbers, change and velocity. The system of Calculus belongs to a stream of thought, and the meaning and application of higher math is found within that stream. To that end, the student is shown the simple geometric meaning of Calculus – being accessible to anyone with a rudimentary knowledge of algebra.
The emphasis of this course is on understanding concepts, and not rehearsing formulas. This class will be especially invaluable to success in college calculus courses where formula is emphasized over understanding.
The goal of this course is to make the student find Calculus enjoyable and simple via an investigation of the historical thoughts which gave rise to its modern form.
Calculus need not be clouded in mystery; by using the power of history and metaphor, this class seeks to shed light on a rather simple subject. Calculus has traditionally been hidden by the mathematicians with their strange symbols and clusters of confusing sentences. This course will reverse that, and show the subject to be quite elementary in every respect.
Course pre-requisites
1. Algebra II, 2. Geometry (or permission of instructor), 3. Pre-Calculus (or permission of instructor).
Required Texts:
Students will be graded on four exams. Retesting is available. When retesting, it is possible to correct previous mistakes and erase past failures. The material covered on each exam comes from lectures, handouts and the two required textbooks. Homework assignments are geared to enforce the material and prepare the students for the exams.
Intro to Math
The Ancient to Modern
Intro to Calculus
I. Math Operators
II. To deliver you from preliminary terrors: symbology of Calculus D and S
III. Different degrees of smallness
IV. y=x^2 graphed and examined with respect to the derivative.
Homework
Reading from Greek Science of the Hellenistic Era
I. Equations: Identity and Conditional
Real numbers and the impossibility of infinity
II. Linear Equations
III. Quadratic Equations
IV. Polynomial Equations of Higher Degrees
V. Division by Zero --> a proof of the horrors coming from it
1. Given the proof of Division by Zero, explain at what step in the proof math is undermined and the whole universe splits apart
2. Working from ax^2 + bx + c = 0, solve for x using all the algebra you can muster. That is, derive the Quadratic Formula on your own. You will need to use the trick of completing the square which is in chapter P of your book.
3. For chapter P section 1, do the following problems: 1, 3, 5, 7, 11, 19, 21, 185, 186, 187, 196. The odd answers are in the back.
I. Inequalities
II. Abs values
III. Solving equations which yield a set of values
I. What is Proof?
II. Use of Metaphor.
III. Greek Mathematics: Understanding Euclid’s Proofs
Chapter P.3
More on Functions and points along lines
The Derivative Again!
Slopes and rate of change
Acceleration and the Slope of a line
Chapter P.4 and graphs of lines
Homework
Take the same assignment from Wed and find the SLOPE between all the points given. What is happening to the slope?
Day 6
Linear equations (Chapter P.5):
Perpendicular slopes
Slope intercept form of lines (y = mx + b)
Graph of | mx + b | = y and how to visualize it
Point slope form of line
y2-y1
-------- = m
x2-x1
Functions (Chapter 1.1)
-------------------------
Understanding y=F(x)
Feeding functions into functions
Domain of Discourse and Range
Out of Range
Homework
Chapter P.5 (5 questions from book)
Day 7
Functions and finding the slope of a tangent line at a given point. The frictionless train example. A full discussion of the point-slope formula as applied to a curve and the derivative.
Day 8
*Test
-- This will cover Chapter P and a bit of chapter 1.
Homework
Hand out article on derivative. Do problems 1-6
Day 8
Overview of Article on the Derivative and Limit from Chapter 1 of Calculus made Easy
Small numbers in the denominator vs. adding small numbers
Homework
Problems 1-6 of Article
Day 9
Finding the Area of a Circle
Introduction to the concept of accounting for small numbers vs my view of erasing small numbers.
Day 10
Functions. Terms, using, and visualizing. Chapter 1.1, 1.2 and 1.3
Homework
Questions from 1.1 and 1.2
p. 75ff: 1,3,15,19 (explain!), 45,85,86,87,[95],[96],[99]
p. 87ff. Calculator problems: 27,29,124
Continue work on problems 1-6 of handout from last week.
Items in [brackets] are problems I stress doing
Day 11
1) A philosophical discussion of infinity and infinitesimals. An introduction to the historical development.
2) Philosophers as theologians in the 16th-19th centuries.
3) God and Infinity.
4) Infinite Series
1 + 1/2 + 1/4 + 1/8 .... a geometric representation
The Dropping Ball Problem 6 + 18/4 + 54/16 + ... + 3/4*(n-1) + ...
5) The Limit and Infinity.
6) The Derivative and Infinity.
Instantaneous Velocity (the Matrix Revolution)
7) The symbol for infinity
8) Infinity divided by zero and the limit of 1/infinity
9) Breaking free from reality:
Calculus as unbounded by empirical constraints:
Assuming infinity when infinity is not experiential.
Homework
Read 3.1 and 3.2 in the text book
We will have a take-home test soon, I mentioned some items that will be on the test (start reviewing):
1) Prove Pythagoreans Theorem geometrically
2) Prove that division by Zero is a
Mathematical Weapon of Mass Destruction
3) Explain what this means in English:
f(x + dx) - f(x)
-------------------
dx
4) Find and Explain the tangent line to a point on a curve.
5) Find a general equation for the slope (m) of a curve
eg. f(x)=3x^2 + 4
I may be so bold as to give
f(x) = x^3, then tell you that m = 3x^2, then ask you to prove it
which would mean finding completing #3 above
given that f(x) = x^3
6) Some basic math problems that I may ask you to work on:
Derive the quadratic formula
using the technique of completing the square
7) Solve some infinite series problems
8) Describe the philosophical dilemma of infinity as you understand it. Take a position in the debate.
9) Be able to enter data into your TI calculator and estimate points on graphs (like x and y intercepts).
10) Discuss the difference between a ratio and a rate of change
Part of the test will be open-book take-home, part of it will be done in class. For example, the Pythagorean proof I will want you to do in class -- same for the Division by zero proof. Note: there are actually two proofs for the Pythagorean Theorem that I presented, I may ask you to give both (the one being rather trivial).
The in-class portion of the test is scheduled for early October.
Day 12
Limits. A Formal Introduction. Chapter 3.2
< I have developed extended notes on the Limit which I will make available >
Homework
Work through all the examples following the formal definition in section 3.2. I can't stress enough the importance of this concept becoming second nature.
Day 13
Limits -> 3.2 and working through examples
Cryptography and using functions to covert passwords into larger numbers not easily reversed.
Homework
3.2 Exercises
Day 14
Limits -> Chapter 3.3
Theorems on Limits: quotient theorem, scalar theorem, etc.
Finding Limits without tripping over zeros. For example, x^3 - 1 / (x - 1) means factoring x^3 - 1 by dividing out the numbers that make it go to zero.
Homework
3.3 Exercises 45,47,49
Day 15
Review material for 2nd Test
Graphic demonstration that the derivative of sine = cosine
Homework
Sample Exam Questions.
Day 16
Limits -> Chapter 3.4
Day 17
Review of problem from test: find derivative of x^3
Cantor and his different infinities. Aleph 0 is the least of infinities. A proof of Aleph 1 and uncountable infinity.
Why sqrt(2) has no home. A number without an address. A proof of why sqrt(2) is homeless and irrational.
Finding the address of all the numbers that are rational. The algorithm: 0/1 1/1 1/0
Continuity, lines, and homeless numbers in a world of non-touching points. Points on a line are all alone in a string of being together, none touching the other, and an infinite number between them.
The Limit and infinity
---------------------
This lesson is designed around the idea of numbers, number lines, infinity, continuity and missing addresses on the number line. Calculus assume continuity, it assumes infinity, and the Limit is the standard way of dealing with it all. The Greeks had a problem with such numbers as sqrt(2), because it did not fit with all the other numbers. It belongs to a class of its own. But what happens as we get close to it along a number line?!
Homework
Read 3.5 on Infinity and the Limit.
Takehome exam over derivatives, infinity, tangents, slopes, rates
Need to know for test: Proof that Sqrt(2) is a Homeless Number
Day 18
Limits to infinity (review w/o working problems)
The sliding ladder problem and rate of change
The derivative again. Review of growing geometric shapes.
Three articles to hand out: 1) how to multiply large numbers in your head, 2) on sqrt(2), and 3) on different infinites (Cantor's proof of uncountable infinity).
Homework
Find the derivative of the sliding ladder y = (181^2 - x^2)^1/2
Find the derivative of x^4 using the method given today in class:
y + dy = (x + dx)^4
Read article on Yo which is a summary of the sqrt(2) proof.
Read article on Transinfinites and Cantor's diagonal proof
Read article on fast-multiply (the quick head-based version!).
Day 19
Section 4.3 and 4.4
Taking the derivative of a constant
proof that it is zero!
Factoring out a constant before taking the derivative
Taking the derivative of sums or differences
Taking the derivative of a product
Taking the derivative of a quotient
Second order and third order derivatives
Homework
Examples from 4.3 (ex. 8 and 9).
Various problems from 4.3 and 4.4 (velocity problem).
Read 4.5 on the chain rule and rework the ladder problem assigned last week: y=(181^2 - x^2)^1/2.
Day 20
Review of assigned problems
Philosophy of Numbers (From Russell's, Mathematical Philosophy)
Paneo's 3 ideas and five rules
The ideas: 0, number, successor
Five rules:
1) 0 is a number
... etc.
What constitutes philosophy of numbers, and what is application. Above and below the line of idea. Being a summary from Russell's discussion of the matter.
The Egyptians and the Pyramid. Egyptian Math and Philosophy as it would be then. The optimal volume of a Pyramid problem. How the Egyptians found the optimal volume without Calculus.
Homework
Learn the three+five items for the next lecture on the subject (they will be on the exam)
Day 21
Setting the first derivative to zero to find the maximum or minimum on a graph. Conceptually, this must be explained by the student in English on the exam.
A) Min -> Max problems using the derivative Intro to 5.7 of textbook
B) Example from secondary text:
1) given two numbers, the sum of them being 10, what can the two number be such that their product is maximum
2) given three numbers, the second twice the first, the sum of all three being 100, find the maximum product of the three.
Homework
Work examples of 5.7 in textbook
Day 22
Application of the Derivative in Min-Max problems
Chapter 5.7
Do example 2 in class
Work first half of problems assigned.
Homework
Various problems from 5.7
Maximize Volume, Area, Profit, etc.
Day 23
Work assigned homework on the board (optimize the area of a semi-circle and rectangle problem).
Quiz (Product Rule, Quotient Rule).
Review of how the Quotient Rule relates to the secant/tangent line method of developing the derivative.
The Chain rule (by example and definition)
Homework
Assigned reading on the Chain Rule.
Day 24
Short review of topic for next test
Chain rule and General Power Rule (4.4 of textbook)
How to program your TI-83 in Basic. We are going to use the calculator to estimate answers that we find with the derivative, so we need to start learning to program it now (first programming exercise comes from my book by Pine).
Homework
Problems from 4.4 -- taking derivatives of more complex functions
Handout on programming the TI-83
Day 25
The implicit and explicit derivative -- the Chain rule in abstract
Working on problem 23 from Chapter 5.7, write a program in BASIC to solve the problem.
Homework
Get the program up and running
Questions from chapter 4.5 on the implicit derivative.
Day 26
Finish work on program. Relate the software we wrote to Calculus and how the two solutions are the same, but the methods are different.
Introduce the IF statement in programming.
Homework
Review for test, finish work from 4.5
Day 27
Explicit vs. Implicit differentiation.
Text next Meeting
Homework
Work on test notes
Day 28
*Test
3 (in-class portion) -- Take home portion assigned
Day 29
Number system memory
Homework
No homework while the students finish their take-home exam.
Day 30
Section 5.1 and 5.2, being max/min finding, critical points, and the meaning of it all. We already did 5.7, which is the application, now I am going back and showing some of the theory behind the application.
Rolle's Theorem and the Mean Value Theorem (it's not really mean, it's quite nice).
Homework
Go back and do Problem 68 from 4.4. The task is to generate the formula for finding the derivative of an absolute value. Quite interesting, and then applicable to section 5.1. In 5.1 do problems 1,5 and 27. For those not in class today, you will need to read 5.1 and 5.2. It's easy stuff, so relax and enjoy the ideas.
Day 31
Chapter 4.5 Again. Implicit and Explicit differentiation. This will be especially needed when we get into applications in chapter 5.
Homework
Questions 6-10 in this section, and Getting the Concepts
Day 32
Review problems from homework; question about (xy)^1/2 -- the trick is that we must use the chain rule then the product rule.
Chapters 5.1 and 5.2
Rolle's Theorem. This time with some history on the man (cf. the book A Tour of Calculus). Rolle rejected Calculus at key points; his Theorem is tool to simplify a lot of ideas into one basic graph/idea.
Mean Value Theorem. This is close to the fundamental theorem of Calculus. We will spend more time on this. I showed how to use the theorem to find the point on an interval with a tangent line parallel to the secant/chord that passes through the endpoints in the interval.
How square root of two is an algebraic number (x^2 - 2 = 0) and how pi is not. I.e. Pi is Transendental. Irrationals and transcendentals, and complex numbers as algebraic (x^2 + 2 = 0). This was all related to finding the critical points on an interval by using the end points and the first derivative. This explains the min-max method in 5.1 and 5.7.
How to map the numbers to letters for memory techniques.
Homework
Questions from 5.2 on the Mean Value Theorem (2-10 and Getting the Concepts).
Day 33
<optional, depending on level of class> Introduction to Trig. Will develop trig tables through rigor and brute force. The idea here is to demystify Trig.
Explain the history of the Mean Value Theorem and its applications.
Day 34
Review of Trig Tables and Calculus and the Derivative.
Homework
Read introductory material in Chapter 10 of the Calc book
Day 35
Retest day. Rework the problems from the tests.
By now, you need to determine which items you want to be retested on.
Day 36
The Limit: Give me an E game
Day 37
1. The Limit -- Rise over Run game.
2. The Area of the Farm problem. Find the least amt of fence.
Homework
Work Limit Chapter: I give you an E, you give me a D problems. These are going to be important in College!
Day 38
1. Trig: Sin(x+y) = Sin(x)Cos(y) + Cos(x)Sin(y)
How can you think of this geometrically, and how can you prove it in terms of the rations. Don't accept formulas, but prove them!
2. Given the volume of a pyramid, find the smallest area to enclose that volume. Then, check to see if the Great Pyramid is close to optimal.
Homework
1) Work on a geometric or algebraic proof of sin() problem.
2) Prove the method of taking the derivative of the area.
Day 39
1) Finish the Pyramid problem.
2) The least area will give you a ration of sqrt(2) for ht to 1/2 base. The Great Pyramid is within 2 degrees of that.
3) Newton-Rhapson. We have been building to finding roots using this method.
Xn+1 = Xn + F(Xn)/ F'(Xn)
Homework
Create your own function, and use the Newton-Rhapson Method to find its roots
Day 40
Review for final.
* Make-up questions.
* Rolle's Theorem. Mean-value theorem
* Trig Tables
* Second Derivative and Inflection Points
* Newton-Rhapson Method
Homework
Handed out make-up exam.
Day 41
Newton-Rhapson Method -- programmed on calculator
Homework
Finish algorithm for Newton Rhapson root finding
Day 42
*
Test 4
Review of in class work. Opportunity to retake will be given on Monday
Day 43
Make-up