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 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.
A Workable (Useable, Useful) Definition
Calculus, its rule and conventions are human inventions. The form of calculus evolved over the years to its present form - how is a good question, beyond the scope of the author.
Earlier, we assumed the slope of a ski located at a point [a,f(a)] on a curve or graph of a function existed and gave the slope of the curve at the same point. That was a quick way to provide motivation for the slope interpretation or sign analysis. But that way does not provide a workable mathematical definition or codification of the concept of what is the slope or derivative of a curve y = f(x) at a point x = a.
The mathematical statement
| m ski = | lim h® 0 |
f(a+h)-f(a) h |
represents our physical expectation that the slope of the secant line should approach the slope m ski of the ski at the point [a,f(a)]/ But as yet we do not have a mathematical definition or codification of what the slope is or should be.
To obtain a starting point for the mathematical theory and treatment of slopes to curves, we codify our physical expectation by declaring the slope m of the curve at y = f(x) to be given by the limit when it exists (that is, when it is given by a finite number) and that the slope provides the slope of a tangent line through the point [x,y] =[a,f(a)].
An illustration follows.(Working) Definition: Assume f(x) denotes a real-value function of a real-valued variable x. If the limit
m = lim
h® 0f(a+h)-f(a)
hexists then the tangent line to the curve curve y = f(x) through [a,f(a)] exists and has slope mtangent = m.
With
= f(a), the linear function
y = m tangent (x-a) + y1where mtangent = mski provides a linear approximation to the value of y = f(x).
Remark (Technical Trap): When we say or write that a limit lim x® c g(x) has an infinite value (or approaches plus or minus infinity), we are describing the behavior of g(x) as x® c but we are not giving a finite number L as the limit. Thus a finite limit does not exist. Now in speaking of limits, mathematics follows the technical convention that a limit
lim x® c g(x)
exists when and only when there is a real number L such that
L = lim x® c g(x) .
In the above definition, the requirement that limit
lim
h® 0f(a+h)-f(a)
h
exist means the quotient
f(a+h)-f(a)
h
tends to a finite value as h® 0.
www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
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.