|
YOU are better than YOU think. Show yourself
how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
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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\
/
\ = /
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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_ / \
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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.
| | Sum Rule
The statement of these rules follows the video examples.
Differentiation rules say how to compute formulas for f '(x1)
in a routine mechanical manner from formulas for f(x), at least when the
formula for f(x) is simple enough. The proof, justification and further
explanation of rules for differentiation may be found in this site area.
Again, the formula or definition
| f¢(x)
= |
lim
h ® 0 |
|
f(x+h) -f (x)
h
|
|
provides the initial limit-based way to compute f '(x). We will use it
to obtain derivatives of simple functions. But we will also introduce
rules of differentiation which permit the calculation of formulas for f '(x)
from formulas for f(x), calculation shortcuts for the evaluation of the limit
definition of f'(x).
Here we see the plan, namely a quantity is represented by a limit. Then rules
are developed to evaluate the limit directly or replace the limit evaluation by
an equivalent calculation in which there is no mention of limits. But the basic
properties of all these calculations come from limit considerations.
Sum Rule: If h(x) = au(x)+ bv(x) for some real numbers a and b,
then g'(x) =a u '(x) + bv'(x)
Assume a = 1 and b = 1 on first reading. The special cases
- a = b =1 gives the sum rule
d ( u + v) = du
+ dv
dx
dx dx
- a = 1 & b = -1 gives the difference rule.
d ( u - v) = du
_ dv
dx
dx dx
Proof:
|
g(x+h)-g(x)
h |
= a |
u(x+h)-u(x)
h |
+ b |
u(x+h)-u(x)
h |
|
Now taking the limit as h approaches zero yields the stated sum rule.
Elementary Example: If f(x) = 4 x + 5 and g(x) = x2
then
f'(x) = 4 and g'(x) = 2x. So h(x) = g(x) + f(x) implies
h'(x) = 2x+4
Generalized
Sum Rule:
| If |
g(x) = |
n
å
j = 1
|
aj* uj(x) |
| then |
g'(x) = |
n
å
j = 1
|
aj* u'j(x) |
The proof follows by mathematical induction on n > 1.
| |
[ Back ] [ Up ] [ Next ]
More Calculus
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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