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.
Triangle Inequality |x +y| < |x| +|y|
The following diagrams illustrate and imply the triangle inequality in for real numbers and for points in the plane.
Geometric View in one dimension
Every real number x and y can be identified with a displacement, arrow, vector or directed line segment.
The length of x + y that is |x+y| equals the sum of the lengths |x| and |y| when x and y have the same direction.
The length of x + y that is |x+y| equals the difference of the lengths |x| and |y| when |x| > |y|.
Therefore |x| > |y|
||x| - |y|| = |x| - |y| < |x +y| < |x| +|y|
Likewise when |y| > |x|
||x| - |y|| = |y| - |x| < |x +y| < |x| +|y|
So in both cases
||x| - |y|| < |x +y| < |x| +|y|
Geometric View in Two Dimensions
Every point x and y in the plane can be identified with an displacement, arrow or vector.
When |x| > |y| and |y| = r > 0, the length L = |x+y| of of x+y satisfies
|x| - r < L < |x| + r
or equivalently
||x| - |y|| = |x| - |y| < L = |x +y| < |x| +|y|
Likewise when |y| > |x|
||x| - |y|| = |y| - |x| < |x +y| < |x| +|y|
So in both cases
||x| - |y|| < |x +y| < |x| +|y|
Remark: The explanation or image in three dimensions is similar:
Remark: When |x| > |y| >0 equality may hold in exactly one of the two inequalities
||x| - |y|| < |x +y| and |x+y| < |x| +|y|
only when x and y are collinear.
Remark: Algebraic proofs dependent on the Pythagorean distance formulas for the distance between points and for the lengths of vectors are available. Students of pure mathematics will or should see them.
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.