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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.
| |
Parameters in Limits
If k is a real number which is constant or does not depend on x, then
limx® a k f(x) = k limx® a f(x)
For example,
limx® 3 k x2 = k limx® 3 x2 = k 32 =
k9 = 9k
when k = 1, 10, 55 or any number you choose.
If a number denoted by a letter k or q or p appears in a limit involving
another (dummy) variable, we call the number and the letter that denotes
it, a parameter.
To learn more about describing and talking about numbers and
quantities. read the long essay [What
is a Variable] and read
Volume 2, Three Skills For Algebra (chapters
8 & 9 ).
A parameter a or x will occur in the forthcoming limit-based introduction of
derivatives.
- lim x® A a x2 +bx+c = a A2 +bA+c
Here a, b, c and A are parameters. The right hand side depends on the
parameters a, b and c, and A. When a =3, b = 4, c =10 and A = 2. The
foregoing equation or template becomes
lim x® 2 3 x2 +4x+10 = 3*22
+4*2+10 = 12 + 8 + 10 = 30
- Let f(x) = x2. Then with the parameter a
= 4 (or another value)
f(a+h) - f(a) = (a+h)2-a2 = a2
+ 2h + h2 = 2ha + h2 =h(2a + h)
Therefore
lim
h® 0 |
f(a+h)-f(a)
h |
= |
lim
h® 0 |
h(2a + h)
h |
= |
lim
h® 0 |
2a + h
|
= 2a
|
Re-read the foregoing with a =3, 8, 99, r (another parameter) or x. What changes?
-
Let f(x) = x2. Now rewrite the foregoing with an
x instead of a.
f(x+h) - f(x) = (x+h)2-x2 = x2
+ 2h + h2 = 2hx + h2 =h(2x + h)
Therefore
lim
h® 0 |
f(x+h)-f(x)
h |
= |
lim
h® 0 |
h(2x + h)
h |
= |
lim
h® 0 |
2x + h
|
= 2x
|
In examples 2 above, the limit process
applied to the a and h dependent expression
eliminates the h dependence and results in an a-dependent
expression 2a. Likewise in example 3, with x in place of the number a in example
2, the evaluation of the limit
results in an expression 2x which depends only on x and not on the eliminated
or dummy variable h. In the limit evaluations above, the variables x and a
are parameters, and values of the limits are parameter dependent.
In the discussion of derivatives for curves or functions y = f(x) in
the next or one of the next lessons, the evaluation of
gives an x-dependent result,
| g(x) = |
lim
h® 0 |
f(x+h)-f(x)
h |
The formula for g(x) also written as f '(x) can be obtained from the
formula for f(x) via a limit-based calculation or via calculation rules,
obtained from the properties of limits, which go directly from the formula for
f(x) to the formula for g(x) without an explicit evaluation of limits.
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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.
[ Back ] [ Up ] [ Next ]
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