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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.
Chapter 4. More Slope Sign Analysis
See the calculus previews 1 and 2 for quick or alternative view of the material in chapters 2 to 5.
Identifying intervals where a slope is positive or negative locates the uphill and downhill portions of a trail y = h(x). Several examples follow. Examples like these require and improve algebraic reasoning skills.
A Linear Function
For the height function y = h(x) = 6x-3, the slope m = 6 is positive everywhere. So on any finite interval [a,b], the height increases as x increases. (Take a = 3 and b = 6 if you wish).
The low-point or least value of the function occurs at the left end x = a. The high point or greatest value occurs at the right end x = b.
Another Linear Function
For the height function y = h(x) = -3x+7, the slope m = -3 is negative everywhere. So on any finite interval [a,b], the height decreases as x increases. (For concreteness, take a = -2 and b = 4 if you wish).
The low-point or least value of the function occurs at the right end x = b. The high point or greatest value occurs at the left end x = a.
[Play Video] 2¼ minutes: Slope Sign Analysis. Example of how to describe where a 2D hill has increasing height and decreasing height from sign analysis of a linear expression for the slope (derivative) of a function.
A Quadratic Function
The slope function (or derivative function) for the quadratic height function
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Solution. The sign analysis follows.
- For x > 3, the factor (x-3) and the slope m = 2(x-3) are both positive.
- For x < 3, the factor (x-3) and the slope m = 2(x-3) are both negative
- For x = 3, the factor (x-3) and the slope m = 2(x-3) are both zero.
The sign analysis leads to the following conclusion. The lowest point on the graph of the quadratic height function y = h(x) = x2-6x+2 is at x = 3.
Note that this conclusion also comes more from a previous knowledge of quadratics. For instance, by completing the square, y = h(x) = x2-6x+2 = (x-3)2-32+2 = (x-3)2-7 ³ -7 with equality only at x = 3. The foregoing sign analysis gives the same information that could have been obtained by another method. In the case of quadratics, sign-analysis of slopes does not give much new information. The calculation of slopes and their sign analysis is of greater interest for more complicated height and slope formulas.
A Cubic
The slope function for the cubic height function
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- the high and lowest points for x in the closed interval [-4,6].
- the greatest and least value of the height h(x) for x in the same interval [-4,6]
Footnote: Factorizations of quadratics can also be done with the help of the quadratic formula. The case where there is no real roots can occur.
Observe
| m = x2-2x-3 = (x-3)(x+1) |
The first two subdiagrams 1 and 2 show the signs of the two factors (x+3) and (x+1) of the slope m = (x+3)(x+1). Subdiagram 3 shows or counts the number of negative signs in the computation of the slope m. This number depends on the factors. Subdiagram 4 shows where the slope m is positive and where it is negative. Based on subdiagram 4, the bottom diagram 5 employ arrows to show where the height y = h(x) is increasing and where it is decreasing. This information then gives or determines the locations of the low and high points in the interval [-4,6] where -4 £ x £ 6.
In particular, observe there are two high points in the interval [-4,6]. One is at x = -1 and the other is at x = 6. It is not possible to say which high point gives the greatest value of h(x) without computing h(-1) and h(6) or otherwise finding the sign of the difference h(6)-h(-1). Now a simple calculation gives h (-1) = 3.67 and h(6) = 20.0. Thus the highest point or peak occurs at x = 6 in this case. There the height is y = 20.0 = h(6). The lowest point in the interval can be found similarly.
[Play Video] 4¼ minutes: Sign Analysis for slope given by product of two linear terms, terms that appear here after the factorization of a quadratic.
Exercises
For each of the following cases where the slope function m is given by a simple formula, find the x coordinate of the high and low points for the corresponding height function y = h(x).
- m = 2 for 10 £ x £ 15
- m = -8 for 2 £ x £ 4
- m = 0 for 1 £ x £ 2.5
- m = x-4 for 0 £ x £ 8
- m = (-1)(x-4) for 0 £ x £ 8
- m = (x-1)(x-2) for 0 £ x £ 4
- m = x2-3x-2 for 0 £ x £ 4
- m = x2+2x+4 for 0 £
x £ 10.
Note that x2+2x+4 = (x+1)2-1+4 = (x+1)2+3 ³ 3 > 0. This quadratic is positive everywhere.
- m = -10(x-1)(x-2) for 0 £ x £ 4
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.