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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
 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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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.

Slope Calculation

[Play Video]  4½  minutes: Approximating Slope of a tangent line, or taking the approximation to Limit, when possible, to give a definition of the slope of a tangent. Saying how to compute or approximate a number or quantity defines.

So far the slope to a curve y = f(x) at a point (x₁,y₁) = (x₁,f(x₁)) has been physically or graphically associated with the slope of a short ski whose midpoint touches a smooth (not too bumpy) curve at the point (x₁,f(x₁)). The following diagram shows or suggests how the slope of such a ski resting on the curve at the point (x₁,y₁) could be approximated by the slope of a short chord joining (x₁,y₁) to a nearby second point (x₂,y₂) = (x₂,f(x₂)) on the curve.

h

The function f(x) is assumed to be continuous at x₁ - without jumps or other discontinuity there.

Consider the following.

1. The chord or line segment joining the point (x₁,y₁) to the point (x₂,y₂) = (x₂,f(x₂) has slope
   

   mchord = y2-y1 x2-x1 = Dy Dx

   and equation y = m_chord(x-x₁)+ y₁. When the ski travels between x = x₁ and x = x₂, its slope is (or should be) approximated by the slope m_chord of the chord, alias line segment.
2. We suppose the point (x₁,y₁) is fixed in place. In other words, suppose it is not moving. We further suppose the point (x₂,y₂) = (x₂,f(x₂)) moves along the curve y = f(x) towards the point (x₁,y₁). The slope m of the line segment through these two points should approach the slope m_ski = f¢(x₁) of a ski on the curve at (x₁,y₁).
3. In the motion just described, as the point (x₂,y₂) = (x₂,f(x₂)) moves along the curve y = f(x) towards the point (x₁,y₁), the abscissa x₂ should move closer to x₁. The difference Dx = x₂-x₁ should also become closer and closer to zero. Thus we expect the approximation 

4. mski » Dy Dx = y2-y1 x2-x1 = f(x2)-f(x1) x2-x1

   to improve when (x₂,y₂) = (x₂,f(x₂)) approaches (x₁,y₁) = (x₁,f(x₁)) and/or as x₂ approaches x₁.

 

5. The continuity of f(x) at x₁ implies the moving point (x₂,y₂) = (x₂,f(x₂)) will approach the non-moving, that is fixed point, (x₁,y₁) = (x₁,f(x₁)) when the abscissa x₂ approaches x₁ or equivalently, when Dx = x₂-x₁ approaches 0.
6. Note the arrow ® will be employed as shorthand for the phrase approaches or goes to.

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.

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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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