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 |
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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.
The Natural Logarithm
For real numbers, the following sections describe the area-under-a-curve definition of the natural logarithm, and how this introduction of the natural logarithm leads to the definition and properties of all logarithms, exponentials and powers involving real numbers.The natural logarithm ln(a) for a > 0 can be introduced
FOOTNOTE: Variants of the exposition given here may be presented less cryptically in other texts. The presentation here is show briefly the approach I would like to see favored in schools. Working through the details of this exposition in its present form could be a subject for discussion in a high school math club. Understanding this section and the next demands or provides a sound command of some mathematics beyond arithmetic.as the (signed) area under the curve y = [1/(s)] from s = 1 to s = a. Equivalently, it may be represented by the signed area under the curve u = [1/(v)] from v = 1 to v = a. This definition does not depend on the labelling of the horizontal and vertical axes. See the next two diagrams.
In the next diagram, the area from s = 1 to s = a > 1 can be approximated by slicing it into n vertical rectangles with the same base size [(a-1)/(n)], and then making this base size smaller by letting n®¥ (that is get larger and larger).
The sum of the area of the resulting rectangles approximates to a single number L with greater and greater accuracy, more decimal places say, as n ®¥. This single limit gives what we call ln(a).
FOOTNOTE: The shorthand n®¥ should be read as n tends to (or goes to) infinity. It is left as an exercise for advance students to write on paper the Riemann sums whose limit is or should be the value L.
For a ³ 1, the value of ln(a) is given by the area from s = 1 to s = a under the curve y = [1/(s)]. Here we take or assume ln(1) = 0. It can be shown that ln(a) ® 0 when when a approaches 1 through values above or greater than 1.
The natural logarithm ln(b) of a number b when 0 < b < 1 is defined next.
For 0 < b < 1, the value of ln(b) is given by (-1) times the area under the curve y = [1/(s)] from s = b to s = 1.
The above two diagram illustrate the arithmetic or area-based definition of the natural logarithm ln(a) or ln(b) in the two mutually exclusive cases a > 1 and 0 < b < 1. These definitions imply that ln(x) ® 0 = ln(1) when x ® 1.
Reading Guide. The rest of this section states and indicates the proofs of two algebraic properties of the natural logarithm. The first proof is easy. The second proof is cryptic - material for advanced students. The next section briefly indicates the relationship between the inverses of the logarithms and exponential functions - more material for advance students. Consult another calculus or analysis text for the missing details.
Proof of Property ln([1/(b)]) = -ln(b) for b > 0.
We will show that 0 = ln(b)+ln([1/(b)]) when b > 0. For this, first consider the case a > 1. In the following diagram
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By symmetry (or reflection across the line y = s), ln(a) = Area(B)+Area(A). Therefore ln(a) = Area(B)+Area(C)
Here A is the rectangle with corners (0,1) and (1/a, 1) while C is the rectangle with corners (1/a,0) and (1,1)
Now by definition -ln([1/(a)]) = Area(B)+Area(C).
Therefore -ln([1/(a)]) = ln(a).This in turn implies ln([1/(a)])+ln(a) = 0.whenever a > 1.
Finally, we conclude ln([1/(b)])+ln(b) = 0 whenever b > 0. This follows by putting a = b if b ³ 1 and by putting a = [1/(b)] if 0 < b < 1.) The latter is equivalent to the property ln([1/(b)]) = -ln(b) which we wanted to show.
Fundamental Property of Logarithms
Next we may derive the fundamental property of logarithms, that is| ln(ab) = ln(b) +ln(a). |
Sketch of A First DemonstrationThis indicates a simple demonstration of the fundamental property for the natural logarithm ln(x) for x > 0. The sketch of an alternative proof follows.1. Divide the interval [1,a] on the s-axis into n ³ 1 segments using the end points si = 1+i·[(a-1)/(n)] where 1 £ i £ n. Each segment has length [(a-1)/(n)].
2. On each segment [si,si+1] construct a rectangle whose top just touches the curve y = [1/(s)] at y = [1/(si)]. The sum Sn of the areas
Aj = yj·(si+1-si) = yi· a-1
nof these rectangles provides an approximation to ln(a) which we assume becomes more accurate as n is made larger.
3. Now the rectangle with base [si,si+1] and height [1/(si)] has the same area as the rectangle with base [bsi,bsi+1] and height [1/(bsi)]. But the rectangles with base segments [bsi,bsi+1] and height [1/(bsi)] approximate the area Sba under the curve y = [1/(s)] from s = b to s = ba. So taking the limit as n ®¥ suggests Sba = ln(a).
4. Drawing a graph suggests or implies Sba = ln(ab) -ln(b). Therefore ln(a) = Sab = ln(ab)-ln(b) as well. So we are done in the first case where a > 1 and b > 0. That is, the area Sba under the curve y = [1/(s)] from s = b to s = ba equals the area under the curve y = [1/(s)] from s = 1 to s = ba minus the areas from s = 1 to s = b.
Now the fundamental property of logarithms, that is ln(ab) = ln(b) +ln(a) holds whenever at least one of the factors a and b is greater than 1 (since addition and multiplication of real numbers is commutative.) Now observe for c > 0 that 0 = ln(1) = ln( [1/(c)] ·c) = ln([1/(c)])+ln(c) since c or its reciprocal must be ³ 1. Hence ln(c) = -ln([1/(c)]). This was shown before with the aid of some diagrams. The latter equality prepares us to treat the sole remaining case where both numbers a and b are between 0 and 1. In this case,
as required. Therefore ln(ab) = ln(a)+ln(b) holds whenever a and b are both positive.
ln(ab)
=
-ln( 1
ab)
=
-ln( 1
a1
b)
=
-[ln( 1
a)+ln( 1
b)]
=
-ln( 1
a) + -ln( 1
b) = ln(a)+ln(b)
Sketch of a Second Demonstration. For a > 0, put G(x) = ln(ax). Then value of G(x) is given by the (signed) area from s = 1 to s = ax under the curve y = [1/(s)]. Observe G(1) = ln(a). The area of region D in the following diagram equals G(x+Dx)-G(x).
The height of the region D is approximately [1/(ax)] and its length is precisely a(x+Dx) - ax = aDx. Therefore
This suggests that
G(x+Dx)-G(x) » Area(D) = 1
ax·aDx = 1
x·Dx Similarly F(x) = ln(x) implies that F¢(x) = [1/(x)]. This implies by the Constant Difference Theorem that
G¢(x) =
lim
Dx® 0
G(x+Dx)-G(x)
Dx= 1
xis constant. To evaluate the constant, observe that
ln(ax)-ln(x) = G(x)-F(x) = d since ln(1) = 0. Thus we conclude ln(ax)-ln(x) = ln(a) or equivalently
d = G(1)-F(1) = ln(a)-ln(1) = ln(a) as required.
ln(ax) = ln(a)+ln(x)
The height of the region D is approximately [1/(ax)] and its length is precisely a(x+Dx) - ax = aDx. Therefore
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Logarithms To Base a > 0
The logarithm to base a > 0 is given by loga(x) = [(ln(x))/(ln(a))] when a ¹ 1. The property ln(ax) = ln(a)+ln(x) now implies logc(ab) = logc(b) +logc(a) holds when a, b and c are all positive real numbers with c ¹ 1. The proof is a simple algebraic exercise. Further note that ln(e) = 1 implies loge(x) = ln(x).
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.