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 |
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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.
Where are functions increasing or decreasing?
Where they constant? Can Interval notation help?
Formal Definitions
In the following, suppose y = f(x) is real-valued function of a real variable x whose domain includes a set S.
- Definition: f is constant on S if and only if
for each pair of real numbers a and b in S, f(a) = f(b)
Example of a function constant on an interval [c,d]
- Definition: f is increasing on S if and only if
for each pair of real numbers a and b in S, a < b implies f(a) <
f(b)
Example of a function strictly increasing on interval [c,d]
- Definition: f is strictly increasing on S if and only
if for each pair of real numbers a and b in S, a < b implies f(a)
< f(b)
Example of a function increasing on interval [c,d] but not strictly increasing as it is constant on two subintervals
- Definition: f is strictly decreasing on S if and only
if for each pair of real numbers a and b in S, a < b implies f(a)
> f(b)
Example of a function strictly decreasing on interval [c,d]
- Definition: f is decreasing on S if and only if
for each pair of real numbers a and b in S, a < b implies f(a) >
f(b)
Example of a function decreasing on interval [c,d] but not strictly increasing as it is constant on a subintervals
In our examples, the set S may be an interval in the domain of f, an interval that might be finite, infinite, semi-finite, and in the finite or semi-finite cases may or may not include end-points.
Four examples (graphs) to illustrate the above definitions follow.
Optional Reading for Quebec Math 436 students.
The first symbol below appear with no explanation in the pair of
textbooks approved for use in English language instruction in
mathematics 436,
We may say a function y = f(x) is increasing on an interval I if and only if for every pair of real numbers a and b in I, a < b implies f(a) < f(b). Or, in with symbols
Here ==> means implies. (The first tome in the pair uses the concepts of implication and definition before the attempted explanation of implications and definitions in second tome. So the order of topics in the two tomes is not sequence in this case and possible in other cases as well. To be brief, a definition describe precisely we hope, what something is, so we can recognize an example when seen. Online chapters 2, 3 and 5 in Volume 2. Three Skills for Algebra explain the concept of implication, and their use in deductive chains or reason to arrive at conclusions and organize knowledge. Read those chapters as soon as possible to understand logic and to develop the precision reading and writing skills needed for this course and for word & studies in general. Good luck. |
means or should be read as for all, for every, for each, what
ever sounds best or appears to be the most appropriate. In practice,
the three phrases are usually interchangeable.
means or should be read as there exists, or more
precisely, there is at least one. Connection Between Monotonicity and Injectivity
A function is strictly monotone if and only if it is strictly increasing or strictly decreasing.
Theorem A: If a real-valued function y = f(x) of a real value x is strictly increasing on its domain then f is one-to-one (injective).
| Example where theorem A applies |
Example where theorem A does not apply |
Proof of A: Now for each y in range y, if a and b denote solutions of the equation f(x) = y then a < b, a = b or a > b. Now f(a) = f(b) implies cases a < b and a > b are not possible. So a = b. Therefore, for each y in range f, the equation f(x) = y has one and only one solution. So the function f is injective on its domain.
The proof of theorem B next, is similar.
Theorem B: If a real-valued function y = f(x) of a real value x is strictly decreasing on its domain then f is one-to-one (injective).
| Example where theorem B applies |
Example where theorem B does not apply |
www.whyslopes.com
Analytic Geometry
Area Entrance
Entrance + Pages Below this page
Pages at Current LevelFNs & Dependency FN With Finite Sets FN Vertical Line Rule FN Infinite Domains FN Sets-Theory FN Interval Notation FN: Sets - Continued (FN) Sets & Relations I (FN) Relations & Sets FN Source Target Domain Range (FN) Injective or Not (FN) Sign & Zero Analysis (FN) Increasing/Decreasing (FN) Extrema FN Numerical Exercises FN Step Sawtooth Abs.Value FN Horizontal Line Rule FN Inverse Functions FN Many Ways to Define
Area Entrance (C) Complex Numbers (FN) What are Functions? (FN) Functions - More SZM: Sign Analysis (L) Lines Summary (P) Polynomials (*,+,-) (Q) Quadratics (D) Simplify Square Roots (T) Unit Circle Trig Conic Sections Links More Links
Pages at Above this Page
Extras: Not all perfect.
Equal Sign Use/Abuse Real Numbers Say More Positive Linear Inequalities Triangle Inequality Absolute Value |x| |x| Eq'ns & Inequalities Rectangular Coords Shortest Path Distance Formulas Add & Multiply Points Polar Coordinates Radians (A) Vectors (A) Coordinate Arithmetic (A) Navigation on Maps (A) Addition Geometrically (A) Rotation (PT) Translations (PT) Dilatations PT: Rotations
Links to Site Pages outside this site area follow - co- and pre- requisites.
Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!