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tutorfinder.com.au
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use or become a tutor at your own risk
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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Algebraic Evaluation of Limits
The repetitive nature of the
examples in this section is deliberate. It emphasizes the
algebraic way of writing and thinking.
- [Play Video]
4½ minutes: Algebraic View of Limits. Example involving sums and
quotients.
- [Play
Video] 2½ minutes: Derivative of a Linear
Expression cx+d via Limits.
- [Play
Video] 2¼ minutes: Derivative as a
Limit of a Quotient. First pass at finding the
derivative or slope of f(x) = x2.
Algebraic View.
- [Play
Video] 2¼ minutes: Second pass at
finding the derivative or slope of f(x) = x2 at
two values of x. Numerical Examples of Limit
Evaluation to suggest a pattern.
- [Play
Video] 3¾ minutes: Third pass at
finding the derivative or slope of f(x) = x2.
Back to the algebraic view and a conclusion.
Remark: Many
students survive high school math courses without mastering
the algebraic way of writing and thinking. But mastery of
the latter is necessary for comprehension of the algebraic
computations and reasoning in calculus. I have known
students, who obtained excellent marks in high school mathematics,
to say that the algebraic way of writing and thinking was
strange to them before taking calculus. The assumption that
students have mastered the algebraic thought cannot yet be
made in a calculus course. Thus examples to help with this
mastery may be needed.
Consider the function y = f(x) = x2. We will compute the
slope, that is the derivative of this function at
x = 2, x = 3, x = 5 and x = a. Look for a pattern in the
following arithmetic computations.
Example 1. Let x = 2. Then with x1 = x = 2 and
x2 = x1+dx = x +dx, we have
Of course 2(2) = 4, but for the sake of pattern recognition
and emphasis, we
keep the arithmetic expression 2(2) to the end
of the calculation.
Now
This implies
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lim
Dx ® 0 |
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Dy
Dx |
= |
lim
Dx ® 0 |
2(2)+(Dx) = 2(2) = 4 |
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Example 2. Let x = 3. Then
Therefore
This implies
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lim
Dx ® 0
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Dy
Dx
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= |
lim
Dx ® 0
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2(3)+(Dx) = 2(3) = 6 |
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Example 3. Let x = 5. Then
Therefore we expe
The last equality suggests that
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lim
Dx ® 0
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Dy
Dx |
= |
lim
Dx ® 0
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2(5)+(Dx) = 2(5) = 10 |
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[Play
Video] 2¼ minutes: Derivative as a
Limit of a Quotient. First pass at finding the
derivative or slope of f(x) = x2.
Algebraic View.
[Play
Video] 2¼ minutes: Second pass at
finding the derivative or slope of f(x) = x2 at
two values of x. Numerical Examples of Limit
Evaluation to suggest a pattern.
[Play
Video] 3¾ minutes: Third pass at
finding the derivative or slope of f(x) = x2.
Back to the algebraic view and a conclusion.
The Common Algebraic Pattern
The three examples follow the same pattern. We
will rewrite the above calculations with the
letter a replacing the numbers 2, 3 and/or 5 above,
to emphasize the pattern. In the rewrite below, note that the role
of a below could be played or assumed by each of the numbers 2,
3 or 5 above, another number or another letter!
Example n. Let x = a. Then as before
Therefore
This implies
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lim
Dx ® 0
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Dy
Dx
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= |
lim
Dx ® 0
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2(a)+(Dx) = 2a |
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(Note that in the limit calculation, the variable a is held constant while
Dx ® 0.)
Now we can replace a in the above pattern by x. This yields
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f¢(x) = |
lim
Dx ® 0
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Dy
Dx
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= ¼ = |
lim
Dx®0
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2x+(Dx) = 2x |
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The ¼ indicates reasoning similar or identical to that has gone
before.
Remark (technical). The ratio Å = [(f(x1+Dx)-f(x1))/(Dx)] is not defined at Dx = 0 as
division by zero is not defined. But
the algebraic manipulations above shows that limDx ® 0Å
does exist (at least for the simple case treated).
[Play
Video] 2¼ minutes: Derivative of x3
algebraically via Limits.
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www.whyslopes.com
Volume 3, Why Slopes and More Math -
Foreword, One Calculus preview and Online Chapters:
(V) signals video (RealPlayer Format) to
watch
Area Entrance & Hub
Foreword
Chapter Descriptions
1. Introduction
2. Calculus Starter Lesson
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Content Guide
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
Online Volume 2, Three Skills
for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key
skills and concepts, those needed in calculus, again to make the hard easier.
A visual understanding of complex
numbers may help - serve as back ground info, in partial fraction
decomposition.
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