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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. |
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Theorem G.1 [Rolle's Theorem] Suppose f(x) is continuous on an interval [a,b] and differentiable when a < x < b. Further suppose f(a) = f(b). Then there exists a point c interior to the interval [a,b] with f¢(c) = 0. Note: saying the point c interior to the interval [a,b] means a < c < b.
Proof of Rolle's Theorem.Note that the function f(x) in the above discussion could have several minima and maxima interior to the interval. At these points the slope or derivative f¢(c) = 0.In general, the extreme value theorem for continuous functions implies there exist at least two points xmin and xmax in the interval [a,b] with the property that f(xmin) £ f(x) £ f(xmax) for every x in the interval [a,b]. In particular, this property implies at the end points that f(xmin) £ f(a) = f(b) £ f(xmax).
If f(xmin) < f(a) = f(b) then a < xmin < b since xmin in [a,b] cannot be an end-point. But then f¢(xmin) = 0 since otherwise, an interior minimum would not occur at xmin. (See the earlier discussion of linear approximation.) Thus the conclusion holds with c = xmin.
Now if f(xmin) = f(a) = f(b) then either f(xmax) > f(a) = f(b) or f(xmax) = f(a) = f(b). In the first case f(xmax) > f(a) implies a < xmax < b and hence f¢(xmax) = 0, else xmax would not yield an interior maximum of f(x). In this first case, the conclusion holds with c = xmin.
Now in the remaining second case f(xmax) = f(b) = f(xmin). In this case, the function f(x) is constant on the interval [a,b]. Thus f¢(c) = 0 for every point c in the interval. The conclusion is obvious.
www.whyslopes.com
Real Analysis - Decimal View
Here are the Appendices from Volume 3, Why Slopes and More Math, Chapters 14 to 19 in Vol 3 are related. Here is a reference for college or university mathematics, electrical engineering and physics.
A. What's Next B. Pigeon Hole Principle B. Bolzano-Weierstrass C1. Triangle Inequality C2. Triangle Inequality C. More T.Inequality D. Sets & Sequences D. Monotone Sequences E. Limits, Properties E Limits & Error Control F. Continuous Functions F. Closed Range Thm F. Intermediate Val. Thm F. Compactness Thm F. Equicontinuity Thm F Extreme Value Thm G. Rolle's Theorem etc G. Mean Val. Thm. G. Constant Difference Thm G. Lipschitz Continuity I PS: One Sided Range Theorems G. Velocity Revisited G. Sufficient Conditions H. Riemann Sums Conv H. Lipschitz Continuity II The site area More Calculus contains a one-sided theorem with proof that should be of interest too.
Vol 1A Logic Postscripts
online only:-Proof by Absurdity alias proof by contradiction
How the demand for consistency supports the law of the excluded middle
Reality versus or with the aid of Imagination
Science, Engineering & Math Students: Have you seen a simpler geometric introduction to complex numbers? ( java applet included) . Can you explain what is a variable without using a symbol? Can you derive trig expression for dot & cross & cosine law from complex number properties? For truth tables and indirect methods of reason, see chapters 19-24 & postscripts in Pattern Based Reason and visit Volume 1A, Pattern Based Reason, striving for objectivity, the empirical challenge & limits.
Vol 1A Postscripts
online onlyProof by Absurdity alias proof by contradiction
How the demand for consistency supports the law of the excluded middle
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