Differential Calculus Preview and Starter Lessons

The online version of Volume 3, Why Slopes and More Math, begins calculus starter lessons, calculus previews, which explain why slopes are studied earlier,  which link slopes to calculus, and which require algebraic skills gradually instead of suddenly.  

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. 

Integral Calculus Preview or Starter Lessons

Limits Revisited

• Limits via Algebra may develop the algebraic way of writing and thinking
  with limits and point the way to a better understand of differentiation.
  
• Chapters 15 to 18 develop further the view that taking the limit of a sequence of
  approximations to geometric or physical quantity - what they might be -  may be taken as a definition of the quantity in the case of convergence. The common theme is as follows: Saying how to calculate a number or amount directly or indirectly (for example as a limit) defines it. 
  
• Chapter 14 takes an pre-modern approach to limits, continuity and convergence. It frames these concepts in terms of decimals, or the decimal representation of numbers, and from numerical analysis, the practical or in-principle question of how to control errors in the decimal calculation of functions and limits.  Earlier zeal for modern mathematics lead to an axiomatic approach in high school and college in which theory made no mention of decimals, but decimal themselves were employed in calculations and in the discussion of approximations.  Here is a correction that is sufficient by itself for most students which may also serve as a base for making the modern decimal free theory more accessible - two birds with one stone. 
  
  Duplicate Material:  Chapter 14 in Volume 3  with its discussion of error control in calculation of y = f(x) at or near a point x = a, in intervals centered at x = a to be almost precise, provides a simpler, old-fashion way to think about limits. That old view is enough (sufficient) for students who will not be specializing in undergraduate mathematics. But that old view also makes the the epsilon-delta e-d view more accessible.  As a student I met the the epsilon-delta e-d view and struggled to understand it, and did so temporarily from time to time.  The decimal perspective of error control and convergence however makes the decimal free view redundant and for those who must have, makes the latter more accessible.
  
• chapter 19 describes the area under a curve definition of the natural logarithm ln(x), derives its properties and defines the exponential as the inverse function to it. The treatment here is simple and geometrically oriented.

More Mathematics: 

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. 

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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
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

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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