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YOU are better than YOU think. Show yourself how:
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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 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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-/[]\- 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. |
Calculus & More Math - Area High LightsIn high school, I wondered why slopes and rates of change were met year after year, without end. The study of calculus provides the why. Online Volumes 2 and 3 represents my first attempt to make the hard easier in calculus. Online postscripts and the newer site Calculus Intro area represent the second attempt and it too refers to this one. Explore the second attempt later. Calculus Preview or ReviewThe following pages prepare for calculus. They put ideas easily understood and repeated first, and in doing provide a clear and easier way to learn or teach.
Logic and Algebra Review
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| First Steps in Logic | First Steps in Algebra | |
| 1.
Introduction 2. Implication Rules 3. Chains of Reason 4. Induction (Longer Chains) 5 Knowledge Islands Indirect Logic methods |
7
Arithmetic Skill Check 8 The Three Skills 9 First Skill What is a Variables 10 Two More Skills 11 Why Shorthand 12 Shorthand Usage 13 What's Next 14_Compound_Interest 15 Linear Equations 16 Painless Proofs 17 Pythagoras |
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What's Next 22. Geometric and Arithmetic Sums 23 Summation Notation 24 Investments, Loans, Pensions - Personal Money Calculations 25 Mathematical Induction and Recursion Proofs, Product Notation, & Factorial Notation |
When I taught calculus, I would give my students, fresh from high school, the arithmetic skill check problems, all of them, in chapter 7. Typically, students who complain the most about the check were the ones that needed that check the most.
More on Chapter 14 in Volume 2. The concept of identifying the backward or indirect use, each time it appears, and that appearance is with most formulas in high school and college, provides a vocal, unifying theme in understanding and developing algebraic skills and concepts. See Chapter 14 for a first example. See the site area on fractions, ratios and proportionality for more examples. The typical proportionality relation imay be first use backwards to find the proportionality constant in it before another forward or backward use.
The algebraic way of writing and reasoning is required at full strength in the first and further weeks of calculus. I would give this Calculus Preview Starter Lesson) & Chapters 2 to 6 in the first weeks to develop algebraic reasoning skills and avoid the first shocks in calculus. But calculus involves the full-strength use of algebra in its decimal free discussion and definition of limits - the epsilon-delta e-d
view of Limits.
| A Decimal
Alternative for the decimal -free epsilon-delta e-d view of Limits |
Derivatives -definition via the limit of approximations. | Differentiation Rules and Integration chapters |
| Start with
chapter 15 or 17. 14 Limits & Error Control (V) 14 Limit of a Funtionn. 14. Limited Error Control 14 Significiant Digits 14 Cauchy Limits 14 Sequence Limits 14 Infinite Decimal Arithmetic via limits. PS: More on Limits |
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What is Slope (V) 15 Slope Calculation (V) 15 Slope, a Limit 15 Tangent Lines 15 Linear Approx., 15 Limits via Algebra (V) 15. Differentiation Rules (V) PS Chain Rule I (V) PS. Chain Rule II (V) PS.Sign Analysis (V) 15 Recap. |
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What is Velocity 17 What is Area 18 Integration 18 Area Calculation 18 Function Definition in 6 Ways 19 Logs & Powers 19 Natural Log. 19 Exponential Fn. |
| Chapter 14 in Volume 3
with its discussion of error control in calculation of y = f(x) at or
near a point x = a, in intervals centered at x = a to be almost
precise, provides a simpler, old-fashion way to think about limits.
That old view is enough (sufficient) for students who will not be
specializing in undergraduate mathematics. But that old view also
makes the the epsilon-delta e-d view more
accessible. As a student I met the the epsilon-delta e-d
view and struggled to understand it, and did so temporarily
from time to time. The decimal perspective of error control and
convergence however makes the decimal free view redundant and for
those who must have, makes the latter more accessible.
More Notes and Advice
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The discussion of limits and functions and differentiation can be algebraically difficult, but the meaning of derivatives, why they are computed, can be seen quickly from a few pictures and diagrams. That being said, the calculus preview and the first chapters 2 to 6 in the right navigation bar tell you where calculus is going. The chapters with their slope sign analysis will also help think algebraically. The essay what is a variable also covers the notion of a constant and a parameter, and it may (almost surely I suspect) fill gaps in your comprehension. The companion online volume Three Skills for Algebra.and this one, together with their postscripts review the arithmetic, logic and algebra you most likely need to do better in calculus.
www.whyslopes.com
Volume 3, Why Slopes and More Math - Preview, starter & further lessons for calculus to ease or avoid algebra shock in instruction & self-instructionForeword, One Calculus preview and Online Chapters: (V) signals video (RealPlayer Format) to watchChapters 2 to 6: offer a very simple preview of calculus and a context for earlier study of slopes and factored polynomials
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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