Calculus, Vector and Complex Numbers

Welcome. Site books Three Skills for Algebra and .Why.Slopes.&.More.Math. may also help.

Starter and Warm-up Lessons (Early Calculus or Late Precalculus Students)

This geometric preview and chapters 2 to 6 in Volume 3, Why Slopes and More Math, give a context for the senior high school level study of slopes and of factored polynomials. The same material may be employed at the start of calculus to make it easier. Calculus asks students to calculate derivatives (slopes for straight lines) and to do sign analysis, that is, to say identify interval where derivatives or slopes are zero, positive or negative. While calculus upto the calculation of derivatives is algebraically challenging, the sign analysis and interpretation as introduced in geometric preview and chapters 2 to 6 is very simple. Moreover, it develops algebraic skills in a way that makes the calculation of derivatives and before that limits, much easier.

Vol 2, Three Skills for Algebra covers many topics in algebra and logic that students starting calculus should have mastered or will have to master sooner or later. Also includes arithmetic review problems to catch common mistakes.

Vol. 3, Why Slopes & More Maths, gives starter lessons for differential and integral calculus.

Chapter 1: High School Math Revisited - last minute preparation for calculus. The aim here is to catch common errors, improve reading skills and revisit some basic concepts in algebra - Most calculus text include a chapter reviewing high school material starting with Functions: The site treatment is comprehensives.

The online version of this chapter, see below, starts before that and points students to Arithmetic Review Problems with Hints of Algebra,

Chapter 2: Limits of Functions - Saying how to calculate a number directly or via the limit of approximations defines it. In the study of derivatives, Limits of approximations are used to provide "official" definitions of slopes to curves y = f(x), velocity, other instantaneous rates of change, and acceleration. Areas and volumes are further defined by the limit of approximations (Riemann sum approximations) and some of these limits may be evaluated via reversal of slope or derivative methods. Watch out for a twist: Limits are said to exist only if discrete or continuous quantities approach a finite limit. Then limits with values +oo (plus infinity) or -oo (negative infinity) are defined but said not to exist because they are not finite. This website re-introduces the self-sufficient decimal viewpoint of limits to make them accessible, or to serve as a stepping stone to the algebraically challenging, epsilon-delta, decimal-free, viewpoint - which almost all do not get.

Chapter 3: Derivatives and Differentiation Rules - (1) Derivatives (the slopes of a function in the preview) at a point are defined or calculated via limits of approximations to what the slope of a tangent line should be. (2) The arithmetic viewpoint is easy to follow. But the dependence of derivatives (slopes) of y =f(x) on the x-coordinate requires the algebraic concept of keeping x constant while an h or dx in a secant approximation to the derivative varies towards to zero in a limiting process. That pattern depends on a full mastery of what is a variable. (3) Next, there is a further twist. Namely, differentiation rules give algebraic methods (justified by limit consideration) for calculating derivatives algebraically. (4) Then rules for differentiating (obtaining slopes) for polynomials, trig functions, logs and/or exponentials alone, or combined as in algebraic expressions or composed follow. Limits and properties of these functions play key role in justifying and implying algebraic rules. Your aim here is to master the algebraic rules, and be well aware of how limits were use to imply those rules. Your aim is to also to master the chain-rule. Differentiation rules will also be used alone and in combination. to develop more rules. Mathematical induction will appear in that development. There may be variation between calculus courses. Some will postpone the discussion of logs and exponentials to later. (5) There may also be a discussion of implicit differentiation - very few further topics depend on it. (6) In this or the next chapter, you may meet several theorems - patterns to apply when certain conditions are met. Your aim is to understand their statement and in that statement be aware of the difference between saying A if B and saying A if and only if B. That is where logic appears.

Chapter 4: Application of Derivatives - The calculus starter lessons are previews of these applications. In the previews, formulas for slopes or derivative functions are given, the application here require their calculation and analysis to locate maximums and minimumss of height and also slope functions. Exercises include graphing functions and identifying interior and end point maximums and minimums of functions or their derivatives. Interior maximums and mininumss of derivatives (slopes) are called inflection points. You will also met first (slope) and second derivative (slope of slope) tests for interior maximums and minimums. Graphing may also involve vertical, horizontal and slanted asymptotes. The calculus preview included the first test. Further application include word max-min problems in which you will define a function y = f(x) and have find its max or min. Here velocity, rates of changes appear as derivatives while acceleration appears as a second derivative.

Chapter 5: Integration: The Riemann sum approximation (whatever that may be) of areas under curves y = f(x) between say x = a and x = b (b>a) in the limit, when the limit exists, leads to a definite integral. The first fundamental theorem gives conditions for the existence of that limit with and without the area interpretation. Then the second fundamental theorem of calculus says how to calculate the limit or definite integral with the aid of functions F(x) whose derivative or slope function is f(x). The net result is a Riemann sum approximation and limt process that yields a definite integral involving f(x) which can be calculated by finding (if you can) an anti-derivative of f(x). In the foregoing envelope, you will meet (a) summation notation for sums, see the algebraic properties of sums, derive those properties via mathematical induction, (b) a finer discussion of the Riemann sum approximation process - the requirement for the common or maximum width of rectangles in the Riemann sum approximation process to tend to zero; and (c) ad hoc antidifferentiation methods for finding from f(x) an anti-derivative F(x) with the property that f(x) is the derivative of F(x). Here all the rules for differentiation are applied in reverse. The introduction of indefinite integrals provides a context for this independent of the interpretation of definite integrals as limits of the Riemann sum approximation process. Most likely, you will meet indefinite integrals and anti-derivatives first, along with algebraic properties inherited from those for differentiation.

Chapter 6. Integration Applications:In essence, the applications consist of identifying physical quantities which can be approximated by and calculated in the limit via Riemann sums and the definite integral representation of the limit. Area under a curve is the application met in the introduction and motivation of the Riemann Sum Approximation and Limit process, the process that leads to a definite integral. This Riemann sum approximation and limit process yields definite integral representations and even definitions for areas between two curves via vertical, horizontal or slanted slicing, volumes of solids via vertical, horizontal or slanted slicing, volumes of solid of revolutions via slicing in planes perpendicular to the axis of revolution in the so called disk or washer methods; volumes of solid of revolutions via slicing into cylindrical shells around the axis of revolution. Here the convergence of the Riemann sum implies what an area or volume should be, and so provides a definition of area or volume for regions in 2D and solids in 3D for which area and volume were not previously defined. Further applications of the Riemann Sum Approximation and Limit process yields formulas (definite integrals) for work, fluid pressure, arc-length, moments, center of mass and so on - quantities need in geometry of engineering and physics - and college level statistics

Calculus Guide

Reference Material: - Light reading for calculus.

Vol 2, Three Skills for Algebra covers many topics in algebra and logic that students starting calculus should have mastered or will have to master sooner or later. Also includes arithmetic review problems to catch common mistakes.

Vol. 3, Why Slopes & More Maths, gives starter lessons for differential and integral calculus.

Suggestion: Read both volumes 2 and 3 before calculus and during it.

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.