1. Introduction to Calculus, A preview

Chapter 1. Introduction

A Calculus Preview

Slopes for the graphs of both linear and nonlinear curves y = f(x) are met in late high school or early college calculus courses along with rules for their calculation. In calculus, slopes are called derivatives. Formulas for slopes are obtained or derived from formulas for curves y = f(x).

A simple geometric interpretation of slopes follows. The graph of a function y = f(x) gives a two-dimensional trail through hills and valleys. A skier in crossing such two or three dimensional hills is aware of the slope of the ground and how this slope changes. The skier in question can tell when or where the uphill and downhill sections are located from the slope of a ski. This represents the first easily visualized physical or geometry interpretation of slopes. Further examples will be given.

Rules for differentiation (slope calculation) give formulas for the slopes of functions y = f(x). In the opposite direction, formulas for functions y = f(x) may in some instances be found by reversing the methods of slope calculation, a process called anti-differentiation or integration. Finding a function f(x) from a knowledge of its slope etc., leads to and justifies common formulas for the perimeters, areas of regions in the plane, the length of curves and the volumes, weights and masses of solids.

Other Books

The following why slopes chapters complement what is usually written in algebra and calculus texts about the calculation of slopes and their geometrical or physical interpretation. Their aim is to explain in a simple way why slope calculation (differentiation rules) and the reversal of the slope calculation process (anti-differentiation rules) are of interest. The rules for differentiation and anti-differentiation are somewhat involved. But it is possible without them to grasp clearly many of the ideas and motivations for slope-related computations.

Most of the material below may fit between the definition of slopes for straight lines in a high school algebra or trig course and the calculation of slopes for nonlinear functions in calculus courses. The remaining material may be read in or along side a first or second course on calculus or read before by gifted students (avid readers) still in school.

Remark: The following texts or others will supply the missing details.

Calculus with Analytic Geometry by D. G. Zill, PWS Publisher, 1985
Calculus of One and Several Variables, by S. L. Salas and E. Hille (John Wiley & Sons 1971 and 1974, ISBN 0-471-00956-3).
Calculus by L. Bers (Holt, Rinehart and Winston 1969, SBN 03-065240-5).

The above books or others on calculus should be in a public library or a school library. Just as two views are better than one, so are two calculus books better than one. When the wording in one is obscure or not readily understood, the slightly different description or ordering of the same topics in the other may clarify matters.This advice applies even to the pages of this book. A break from reading might also have the same effect.

Remark: The formal or proper presentation of mathematics requires no diagrams and no physical interpretation/reasoning. But without diagrams and without geometric or physical interpretations in examples, mathematical ideas can be without motivation. The following pages put the motivation first.

Complex Numbers

The chapters on vectors and complex numbers appear after the last why slope chapters, but they may be read before. Some of the chapters in this book are like clothes in a suitcase: the order in which they are packed or unpacked is often unimportant.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

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Chapter 1. Introduction

A Calculus Preview

Other Books

Complex Numbers

Skill Development Tips For All

Skill Development Tips
For All