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| First
Preview (V) 1. Introduction 2 Second Preview Begins (V) 2 Skier in Motion (V) 2 The Skier (V) 2. Position Dependent (V) |
3
Slope Sign Analysis (V) 4 Single Factor Analysis (V) 4 Two Factor (V) 4 More Factors (V) 4 With Divisors (V) 5 Max-Min Tests |
6
Discontinuities 11 Slope of Slope 13 Acceleration |
| November 2008: Flash Video lesson have been included with these chapters. The (V) indicates the presence of earlier recorded, Realplayer format videos. | ||
Calculus mastery requires the algebraic way of writing and reasoning at full strength. The calculus previews below require a knowledge of slopes and polynomials. The previews provide a context for the study of slopes and for the study of the slope-related concept of derivative that appears in calculus. The first previews in visually and geometrically explaining the role of slopes and derivatives in describing functions are increasing and decreasing explain why slopes and derivatives are calculated for curves y = f(x). In the second preview, sign analysis of factored polynomials further develops algebraic reasoning skills by taking an easier part of calculus, what is met after several weeks of study, and putting it earlier. One or both previews can be met and enjoyed as motivation for technical elements of secondary mathematics in pre-calculus class which explain the role of slope for straight lines y = mx + b, and beyond that or beside describe quadratics and the factorization of polynomials.
A Twist to be Aware of: Slopes of a curve y = f(x) (derivatives of f(x)) are approximated by slopes of secant lines and defined as the limit of these approximations. That provides the first view of what is a slope or derivative. BUT properties of limits imply rules for obtaining derivatives which depend on the algebraic form of a function f(x), rules in which limits are not seen, albeit limits are beneath the surface in that they implied the rules: Those rules include sum, difference, product, quotient and chain rules plus all the rules for differentiating basic functions: trig, polynomial, exponential, logarithmic. In sum, the derivatives are formally introduced as limits of slopes to secants - the slope of a ski viewpoint in the starter lessons are too informal. But rules for evaluation these limits (rules based on the properties of limits, etc) appear to give formulas for the calculation of derivatives without mention of derivatives.
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17 What
is Area 18 Integration 18 Area Calculation |
Chapter 17 and 18 offer a context for the
discussion of areas under curves plus statements of the
fundamental theorem of calculus. All this is done without
reference to summation notation for Riemann Sums
This summation notation free approach provides tutors and teachers a simpler route for defining the definite integral as limit of Riemann sums approximations to what area should be. |
| A Decimal
Alternative for the decimal -free epsilon-delta e-d view of Limits |
Derivatives -definition via the limit of approximations. | Differentiation Rules and Integration chapters |
| Start with
chapter 15 or 17. 14 Limits & Error Control (V) 14 Limit of a Funtionn. 14. Limited Error Control 14 Significiant Digits 14 Cauchy Limits 14 Sequence Limits 14 Infinite Decimal Arithmetic via limits. PS: More on Limits |
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. Differentiation Rules (V) PS Chain Rule I (V) PS. Chain Rule II (V) PS.Sign Analysis (V) 15 Recap. |
16
What is Velocity 17 What is Area 18 Integration 18 Area Calculation 19 Logs & Powers 19 Natural Log. 19 Exponential Fn. |
Chapters 21 to 24 struggle with vectors and an introduction to complex numbers. I, the site and book author, saw the late Richard Feynmann in 20 minutes of three evenings of 1979 public lectures at McGill, describe his discipline as a the multiplication of arrows in the plane using a rule add the angles, multiple the lengths. Since then I have wondered how to incorporate that view of complex number, Feynman never mention in his presentation to a general audience, into the exposition of high school mathematics and calculus. Chapters 22 and 23 do so with the help of assumptions above rigid body motions - a concept from Euclidean Geometry.
Since the 1995 composition of chapters 22 and 23, site pages have explored the easy consequences, the equalities that follow, from having two different ways of calculating products of complex numbers, and site pages have explored different means for developing complex numbers and their properties from Euclidean geometry and an applied mathematics (Euclidean Geometry "justified) assumptions about locating points with rectangular and polar coordinates, all in the pre-modern style met in the high school exposition of trigonometry and/or Euclidean Geometry itself. You will find developments in full or part of complex numbers in 7. Analytic Geometry/Functions 2006 8. Number Theory. 2006-7 and 9. Complex Numbers More 2001, in top level page complex numbers, and most recently in the an -Euclidean-Geometry extended in summer 2008 to include a treatment of complex numbers which in retrospect reproduces in disguised but more comfortable form the rigid body arguments in chapters 22 and 23 that implied the distributive law for multiplication over addition of arrows in the plane. That ends a journey that began in 1979 with a lecture of Richard Feymann.
The struggle with vectors met in chapters 21 and 22 reflects a conflict between pure mathematics and the assumptions necessary in applied mathematics to work with coordinates. That issue is resolved or will be resolved in an applied mathematics manner in a future growth of the analytic geometry site area - we hope.
Why Slopes
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If you like Volume 3 you may also like Three Skills for Algebra , Exponents & Radicals Exactly, complex numbers, Euclidean Geometry , More Calculus and Pattern Based Reason as well. |
Units in Calculations:
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
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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