Calculus Introduction

Appetizers and Lessons for Mathematics and Reason ( Français)
www.whyslopes.com [ Back ] [ Up ]
Logic mastery is key to easing or avoiding learning difficulties in work & studies.

Chapter 5 - Ideas for Calculus

Starter Lessons or Methods to make calculus easier

This site area does not offer a complete skill and concept development path for calculus. Instead it present several ideas to make a first course in calculus more accessible. Later, site expansion may lead to more starter lessons and key phrases to make skill and concept development more consistent and simpler.

This page links to existing site material in calculus. Experienced calculus teachers will see in Volumes 2 and 3, and in the More Calculus site areas an exploration of methods to prepare students for calculus and to change the order of topics in calculus to make the hard easier. Some comment will follow later on course design and preparation. In the mean-time, for instruction and self-instruction, readers will have to explore existing site material to identify some, if not all the key themes and concepts that will appear in this chapter on calculus instruction and self-instruction.

Volume 3 in chapters 2 to 6, and 11 to 18, provides ideas to make the introduction of differential and integral calculus more accessible. Volume 2 in its coverage of logic may point students to greater precision in reading and writing in the first instance and a better understanding of the logic met in calculus explanations and applications in the second instance. Volume 2 coverage of three (correction four) four skills for algebra and the Volume 2 postscript on what is a variable will provide students with a clearer understanding of the algebraic way of writing and reasoning, all in preparation for calculus. The development of summation notation with formulas for the evaluation of arithmetic and geometric sums represents further preparation for calculus (integral calculus) in Volume 2.

The definition of derivatives, velocity, acceleration and area under a curve all follow the same twist. Saying how to compute a number directly or via limits defines it. There lies a theme for the use of limits in defining slope to tangent lines, derivatives, velocity, acceleration and areas, etc.

The study of calculus explains why slopes appear in secondary school mathematics, year after year. Calculus provides a context and a reason for many components of secondary mathematics: functions, trig, analytic and Euclidean Geometry, logic and arithmetic.

Calculus in the first instance consists of slope related calculations, their interpretation and reversal. Calculus provides a language for discussing numbers and quantities, and the relations between them in accounting, investing, engineering and science.

The following "chapters" link to material in site area to make, we hope, the hard easier to learn and teacher.

Chapter 0: Two Calculus Previews - Calculus requires the algebraic way of writing and reasoning suddenly and at full strength. These previews readable in pre-calculus courses, give a context for the study of slopes in high school and a way to ease or avoid difficulties.

Remark: The geometric and algebraic previews of calculus provide motivation for the calculation of slopes (derivatives) for nonlinear functions and put first the relatively easy sign analysis of derivatives, derivatives that students are given instead of being asked to derive, to develop their algebraic reasoning skills. Is possible that exercises like appeared in a version of the modern mathematics curricula, I seen hints of that, but no concrete evidence. That being said, these preview could appear along side the introduction and discussion of polynomial and rational functions to provide motivation for the latter.

Chapter 1: Preparation for Calculus (Arithmetic Review Problems with Hints of Algebra, Algebra Notes, Logic). The aim here is to catch common errors, improve reading skills and revisit some basic concepts in algebra - High School level material. Most calculus text include a chapter reviewing high school material. New: Animated Examples illustrate some skills and concepts.

Remark: This chapter is needed by students who have not be subject to a demanding drill and practice in the evaluation or simplification of arithmetic expressions.

Chapter 2: All About Limits - Motivation, Numerical Evaluation, Algebraic Evaluation, How Continuity Permits Evaluation by Substitution. Limits are employed in calculus and its applications to define key number or quantities - saying how to compute a number in the limit via a sequence of approximations defines it. After the definition, properties of limits may lead to rules for obtaining the number in question algebraically. New: Animated Examples illustrate limit calculations.

Keeping Up Appearances: Master the differentiation rules and there uses first, and leave the technical explanation to later.

Remark: The decimal viewpoint of real numbers and of error control in the evaluation of functions at points where the latter are continuous, and error control in the evaluation of limits, those that converge, is sufficient or more than sufficient for students before they enter, if at all, the study of pure mathematics.

Chapter 3: Derivatives - Introduction and Calculation. The calculus preview provides motivation for the discussion of derivatives - the approximation of what they should be, and then a definition using the limit of approximation (should that limit be defined). Saying how to compute a number ( here slope to tangent line) directly or via limits defines it. Then differentiation rules give methods for evaluating limits algebraically without mention of the limits. The application of rules for differentiation hides the presence of limit consideration, but limit consideration are present in the derivation of those rules. New: This month, March 2006, animated examples are being added to illustrate the differentiation rules.

Keeping Up Appearances: Master the differentiation rules and there uses first, and leave the technical explanation to later. Give priority to those technical explanation met in class.

Chapter 4: More on Limits and Derivatives - Notes on Application of First Derivative, Hint of Second Derivative. Velocity as a Limit, - Saying how to compute a number directly or via limits defines it. Animated Examples are being added.

Chapter 5: Area and Integrals: Introduction of Area via Limits (Saying how to compute an area directly or via limits defines it), Calculation with Anti-Derivatives. Animated Examples to Come

Extra Material: Theorems and Proofs : Here the proofs and concepts normally omitted or not seen in first and further courses in calculus. The treatment here provide a simpler, but not a simple path through the proofs with a few variations - pointers to an alternative calculus program & a context for ideas that gifted and talented students, or students who insist on having proofs may appreciate. The One Sided Range Theorems appear to be site Eurekas - a publishable paragraph perhaps. First time readers should scan the theorems and skip the proofs on first reading.

Professor Whyslopes' mistake as student was to refuse to use a formula or method until he understood its justification in full. He should have learnt to use it first for the sake of appearances or marks, and leave comprehension of challenging material to later or holidays.