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.
From Slope Approximation
to a working Definition
Saying precisely how to compute a quantity, defines it.
Motivation for Limit Definition or Codification of Derivatives
Note sure that is clear enough, a rewrite or elimination in order
For straight lines, slopes can be defined by a simple formula
| The slope m of a straight line segment between two points (x1,y1)
and (x2,y2) may be calculated as
follows.
|
But for curves y = f(x), we can approximate what we think the slope should be at point (x1,y1) on the graph of y = f(x), and see whether or not the approximation get closer and closer to a single number m as the the approximations get better.
| The following diagram shows or suggests how the slope
of such a ski resting on the curve at the point (x1,y1)
could be approximated by the slope of a short chord joining (x1,y1)
to a nearby second point (x2,y2) = (x2,
f(x2) ) on the curve.
If the approximations converge to a finite value, we say the limiting value of the slope of the secants (line segments) between (x1,y1) and nearby points (x2,y2) is the derivative or slope of the function y = f(x) at x = x1. |
Here the slope of the secant is
| slope m = | Dy
Dx |
= | y2-y1
x2-x1 |
= | rise
run |
So we take the limit of approximations, if it exists, to be the slope or derivative f '(x1) of a function or curve y = f(x) at at x = x1. That is
| f '(x1) | = |
lim Dx ® 0 |
Dy
Dx |
||
| = |
lim Dx ® 0 |
f(x1+Dx) -f
(x1)
Dx |
In mathematics, saying how to compute a number or quantity directly or via a convergent process (a limit), defines it. In calculus, first slopes or derivatives and later area of regions are defined using limits. So we need to understand the theory or properties of limits. The question becomes what is a limit and how do with obtain their values. So we are going to cover two simplest view of limits. A more complicated view is left for later.
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In general there is no direct algebraic definition of what is the slope at a point x = a in the domain of a curve or function y =f(x). But we can approximate the slope algebraically by computing the slope of a shorter and shorter secant chord between a pair of point (x,y) = (a, f(a)) and (X,Y) = (a+h, f(a+h) on the graph. The limiting value of the slope of the secant as h tends to 0 is taken to be the slope of the curve at x = a. So in the first instance, we use a limit-based definition of the slope m at a single point x= a. The process is called differentiation. The value of the slope m is obtained or derived from the formula or function y =f (x). That may justify calling the slope m at x = a the derivative of function f(x) at x = a.
Now the slope m computed gives a value dependent on the location x = a. So the slope m = g(a) for some function g(x). Properties of limits lead to algebraic rules for hiding the limit-based calculation of slope m = g(a) by providing limit-free rules for obtaining or deriving a formula for the slope function g(x) from formulas for the curve height function. f(x). That may justify calling g(x), the derivative of function f(x) and writing g(x) =f'(x) to indicate that g(x) is derived from f(x).
This theory to practice pattern of defining or introducing a number via limit-based considerations (approximations) and then seeking algebraically simpler ways to evaluate the limit is repeated in calculus. |
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.