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A Calculus, Preparation for Calculus and Math Ed ReformWebsite, Etc.

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1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math
(a calculus preview/review)

Mathematics Course Designers: LAMP offers food for thought.
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||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||
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YOU are better than YOU think. Show yourself  how:  

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Read  logic chapters 1 to 5  in online volume Three Skills for Algebra  for greater skills & confidence in  work 
and study

Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention.

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.


After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.


Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.


Decimal Insights on Limits, Continuity, Convergence

Decimal and decimal-free error-control perspectives of continuity, limits and Cauchy sequences are given below. These perspectives is followed by comments on math education.

Continuity and Unlimited Error Control

Limits and continuity in calculus may be described geometrically, that is, intuitively and informally, or more precisely in terms of say epsilons and deltas. The roles of epsilon and delta below are played by E > 0 and D > 0.

Imagine for instance we want to compute a function f(x) at the point x = A accurately. So we can ask the error control question how close must x be to A in order for f(x) to agree with f(A) to say k-decimal places. The answer might be that x must agree with A to m decimal places. In some computational problems, this answer for a specified number k of decimal places may be all that is needed. But in other situations, we want in practice or in principle, unlimited error control. Here we may want to say for any k, there is an m such f(x) will agree with f(A) to k decimals if x agrees with A to m decimals. Unlimited error control offers motivation and a perspective on the discussion of continuity.

Now will say that f(x) is continuous at x = A if for each whole number k, there is a number m such the limit f(a) and the value of f(x) will agree to k-decimals whenever the number x agrees with the value of a to m decimal places. Continuity here represents the concept of unlimited error control in decimal computations.

More generally, we can ask (following Cauchy), given an error control target E > 0, how close must x be to A for the difference of f(x) and f(A) to be less than E in magnitude? The answer follows by obtaining a number D with the property that if |x-A| < d then |f(x) - f(A)| < E.

Without reference to decimals we can say that f(x) is continuous at x = A if for every error control tolerance E > 0, there is a number D > 0 such that whenever |x-A| < d then |f(x)- f(A)| < E. Here continuity at x="A" corresponds to the idea of unlimited error control at x="A." This second concept is decimal free. It is traditional to use epsilons and deltas in place of E> 0 and D > 0.

Limits

We will say that a number L is the limit of a function f(x) as x approaches A if one of the following conditions hold:

  • (Decimal Perspective): For every whole number k, there exist a whole number m such f(x) will agree will L to k decimal places if x agrees with A to m decimal places.
  • (Decimal Free Perspective): For every positive number E > 0, there exists a positive number D> 0 such that if |x-A| < d then |f(x)- f(A)| < E.

Both conditions are equivalent. Each implies the other. Why or how depends on how you think of (or represent) the real numbers. For most people, assuming that real numbers are represented by signed decimal expansions (infinite or finite) is sufficient. Modern mathematics has alternate decimal (or base) -free representations of real numbers.


 

 

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More Calculus
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Calculus Videos
0.   First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn:  Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
3. Derivative Motivation
3.  Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4.  Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5  Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere

To Learn More, visit Volumes 2 and 3.


Advanced Topics

Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity

To Learn More: Visit Real Analysis.

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