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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.
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Range and Intermediate Value Theorems
In the previous lesson, we proved the following.
One Sided Range Theorem: Let I be a non-empty interval. Suppose f: I
--> R is an equi-continuous function or Lipshitz Continuous function on I.
Suppose M is real number not in the range f(I) of f and not a limit
point of f(I). Then either (i) for all x in I, f(x) < M or (ii)
for all x in I, f(x) > M.
Contrapositive Version: If M is a real number for which there
are points a and b with f(a) < M < f(b) then M is a limit
point of f(I) or M is in the image f(I).
Intermediate Value Theorem: Let I be a finite interval [a,b]. Suppose
f: I --> R is an equi-continuous function or Lipshitz Continuous function on
[a,b]. If a real number M satisfies f(c) < M <
f(d) for some points c and d in I then M belongs to the range of f and M =
f(q) for some point q in the interval [a,b]..
Proof: The Contrapositive version of the one side range theorem
implies M is the image f([a,b]), in which case there is nothing to more to
show, or M is a limit point of the image f([a,b]). But the image of a closed
interval [a,b] under a continuous function is closed. That is, it contains all
its limit points. So again M is in the image f([a,b]).
If I is a finite interval [a, b] then continuity
of f(x) on I implies equicontinuity on I. That implies the following.
Usual Intermediate Value Theorem: Suppose f: [a, b] -->
R is an continuous. Suppose for some numbers a < b in I, the real
number y belongs to the interval with end points f(a) and f(b). Then there
a least one point x in the interval with endpoints a and b such that f(c) =
y.
Extreme Value Theorem Revisited. Suppose b > a.
Suppose f(x) is continuous at each point in the interval [a,b].
Further, there exist a unique pair of real numbers ymin
and ymax such that f([a,b]) = [ymin,
ymax].
In other words, the image of a closed interval [a, b] under a continuous,
equi-continous or Lipshitz continuous functions is a closed interval.
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www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
Up
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.
[ Back ] [ Up ] [ Next ]
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