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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. |
Chapter 14
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The Greek letters d (delta) and e (epsilon) above are employed here in accordance with tradition of some (not all) calculus texts. For simplicity, the error control tolerances e and d in the first instance here and below, may be restricted to be numbers of the form [1/2] ·10-k = [1/2] [1/(10k)]. The decimal free discussion of error control and continuity dispenses with this requirement.
We say a function f(x) is continuous at a point x = a if and only if unlimited error control is possible there. More formally, we state the following definition.
Theorem 14.1 [Continuity at a Point] If f(x) is a real-valued function of a real number x in an interval [c,d], and a is a number in the interval [c,d] then the function f is said to be continuous at the number x = a if and only if the following holds. If for every n, there exist an m such that
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[Continuity at Point] If f(x) is a real-valued function of a real number x in an interval [c,d], and a is a point in the interval [c,d] then the function f is said to be continuous at x = a if and only if the following holds: For every e1 > 0, there exist a d1 > 0 such that
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Proof of Equivalence.To show the decimal-based description implies the decimal-free description of continuity, observe the following. First given e1 > 0, there is an n > 0 such that e1 > [1/2]·[1/(10n)] = e. The decimal-based requirement for continuity now is satisfied for some d = [1/2] ·[1/(10m)]. So the decimal-free version holds with d1 = d = [1/2]·[1/(10m)].
Conversely, the other way that is, to show the latter decimal-free form implies the decimal-based description of continuity, observe the following. Given m > 0, let e1 = e = [1/2] ·[1/(10m)]. Then choose d1 > 0 so that the decimal-free requirement is satisfied. The decimal-based version is then satisfied if m > 0 is selected so that d1 ³ d = [1/2] ·[1/(10m)].
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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