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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.
| |
Limits Evaluation by immediate or delayed substitution
Immediate substitution of x = a into an expression f(x)
is possible when and only when a the expression or function f(x) is
continuous at x = a.
Limits, Algebraic Evaluation
Here are some limits we are going to evaluate algebraically using the
algebraic described properties of limits in the previous lesson.
A = lim x® 2
3x+4
B = lim x®
6 5 x2-
8x
C = lim x® -2
4x3-3x+ 1
D = lim y ®
-2 4y3-3y+ 1
and
Solutions
For the calculation
A = lim x® 2
3x+4
as x ® 2 we observe or assume
3x ® 6 and so observe or assume 3x+4 ®
10. We can also write more briefly
A = lim x® 2
3x+4 = 6+ 4 = 10 (result)
Second
B = lim x® 6
5 x2- 8x
= 5(6)2 - 8(6)
= 5(30) - 48
= 150 - 48 = 102 (result)
| The limit evaluation process x®
6 applied to 5 x2- 8x, an expression
dependent on x, results in a number -25 which does not
depend on x. This limit evaluation process eliminates the x
dependence. When we apply a limit process to an formula or
function f(x) which eliminates the x-dependence, we call the
letter or placeholder x, a dummy variable. |
Third
C = lim x® -2 4x3-3x+
1
= 4 (-2)3 - 3(-2) + 1
= 4(-8) +6 + 1 = -32+ 7
= -25 (result)
Fourth we can evaluate
D = lim y ® -2
4y3-3y+ 1 = -25
directly by same reasoning we did for C. Simply replace the x by a y.
| More on Dummy Variables: The letters x and
y in the expressions for C and D have the same roles. In the
expressions x® -2 and y ®
-2 they both represent the ideas of a number approaching the value
-2. But the results for C and D do not depend on our choice of
letters in the limit expressions for them.
In the evaluation of a limit
the value L of the limit does not depend on x, limit
evaluation eliminates that dependence, but the value of L may
depend on a. |
Fifth, for the evaluation of the limit
we observe the attempt to evaluate inside expression
by the immediate substitution of 5 for x (x =5) in the yields 0/0, a
fraction with a zero in the denominator, a fraction which has no numerical
definition or value. It is undefined. But observe the inside expression
|
f(x) = |
x-5
x2-5x |
= |
x-5
(x-5)x |
= |
1
x |
® |
1
5 |
when x ® 5 |
The foregoing suggests the values of the inside expression
f(x) =
x-5
x2-5x
approach 1/5 or 0.2 as x approaches 5 even thought f(0) is not defined.
Note: The nuance, subtlely or technicality here is that in
the evaluation of a limit
the value of the inside expression f(x) at the limiting value x = a of
x as x approaches a is not of interest. It does not have to be defined.
Limit evaluation here is independent of what happens at x = a. Limit
evaluation is based gives the limiting value of f(x) when x is restricted
to smaller and smaller a-deleted intervals centred at x = a, that is
intervals in which the value a has been removed. |
The foregoing discussion needs to be understood, but when we evaluate the
limit
we write less. In particular we write
|
E = |
lim
x® 5 |
x-5
x2-5x | |
= |
lim
x® 5 |
x-5
(x-5)x | |
= |
lim
x® 5 |
1
x | |
= |
1
5 |
(exact
answer) |
The avoidance of 0/0 by replacing the initial expression by another
represents a delay substitution.
Please leave your results as a fraction. Here
the answer can be expressed exactly as a decimal but in correcting your
results and any work leading to your results it easier to recognize a fraction
(reduced to lowest forms) than it is to recognize a decimal. So do exact
arithmetic with fractions and radicals (square roots etc instead of using your
calculator. This instructionto do exact arithmetic and avoid decimals,
or postpone their use until all possible exact arithmetic is done with whole
numbers and fractions provides a standard to meet in your mastery of calculus.
I have done the limit evaluation process over several lines. You could do it
in one lines.
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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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