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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 13
Acceleration
Acceleration As a Slope of A Slope
A graph d = f(t) of distance versus time is shown in the first
graph. The second graph shows the graph of the velocity v = f¢(t),
that is, the slope of the first (distance) graph, versus
time. The third graph maps the slope a = f¢¢(t) of the
second (velocity) graph versus time. This second slope is
called the acceleration.
FOOTNOTE: The word derivative
could and perhaps should be used here in place of the
word slope.
At time t1, the acceleration is greatest. The rate of
change of velocity before and after is less. At time t2,
the velocity changes sign and the direction of travel
changes as well from negative to positive. The distance to the
origin is a minimum at t2.
Our physical sense of speed and velocity corresponds to the
measurement of slope and the behavior of this slope on a
distance versus time graph. Thus driving or riding on a
bicycle or in a car, our physical senses
indicate when the speed is changing in magnitude or
direction. In a car or on a bicycle, a driver or rider
can sense where the speed or velocity
As slope or velocity increases, the forward motion accelerates. When
velocity decreases the forward motion decelerates. In
car or bicycle motion, acceleration comes from applying
enough energy to overcome rolling resistance while
deceleration comes from a lack of such energy or from deliberate
braking.
Another Varying Velocity Example
The direction of travel changes and the sign of the slope,
here velocity, changes at times t = t2 and t = t4. The
slope or velocity is greatest at the times
t = t1 and least at the t = t3. These two times
yield inflection points on the distance versus time graph.
These points identify the
times when the speed (= the magnitude of the slope on the
distance versus time graph) is
greatest and least.
Varying UniDirectional Velocity
The first diagram above shows the graph of distance versus time. The slope of this first graph versus time is plotted in the second diagram. The slope with its units of distance over time has the previously mentioned interpretation of speed or velocity. Here the speed (slope) in this graph remains positive, but it increases and decreases. See the second graph. Physically, this corresponds to a unidirectional, forward motion which is subject to acceleration and deceleration, that is speed increases and decreases.
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.