YOU are better than YOU think. Show yourself  how:  

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 Logic chapters 1 to 5  re- appear not in sequence, as is or longer,  in  Volume 1A,  Pattern Based Reason, Bon Appetite.

Logic Mastery
 Amazing, Amusing, Amorous,  Delicious, Delightful, Edifying, Strengthening Elixir. 
It eases work & learning difficulties Makes the hard easier. Opens eyes. Leads to greater precision.
in reading and
writing

Logic mastery makes the hard, easier. Logic mastery  leads to better, stronger and richer comprehension.  Logic mastery  improves reading and writing.  Logic mastery ease learning difficulties.  Logic mastery gives a headstart.  In sum, logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

After logic,   (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;

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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.

Try the Twiddla Whiteboard. In principle, it  allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio.  Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice.

For online automated help in senior high school maths & calculus, visit  quickmath.com  For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com  With  overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck.

The Volume 3 site area and this Volume 3, extras and postscripts, site area provide overlapping guides to calculus.  Student planning to take calculus or just starting it should see the geometric and algebraic calculus previews and starter lessons in the Volume 3 site area. Then there is a choice of reading more of Volume 3 and exploring this site area.

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 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. Do not delay. Follow the advice and directions below in the hope of easing or avoiding difficulties. The following "chapters" link to material in this site area and across the rest of this site to make, we hope, the hard easier and/or to identify what you need to master alone or with help.  

• 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. See if it works.

• 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.

• 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. Give priority to those technical explanation met in class.

• 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). Then differentiation rules give methods for evaluating limits algebraically without mention the limits. 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 - Velocity as a Limit, Notes on Application of First Derivative, Hint of Second Derivative.  - Saying how to compute a number directly or via limits defines it.  Animated Examples are being added.

• Chapter 5: Area and Integrals: Introduction via Limits and 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. 

For a second course in calculus, the site introduction of complex numbers may be useful. 

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.

Page Content Begins

Chapter 0. Calculus Previews 

1. This Calculus Preview gives a first image of calculus - to explain why derivatives are calculated and how they are used, and to give a context for earlier studies of slopes and rates of change. 
   

2. Geometric preview in a more algebraic manner. The first lesson should not contain anything new. 

   Algebra Preview:

   2 Slopes Revisited (V)
   2 Skier in Motion (V)
   2 The Skier (V)
   2. Position Dependent (V)
   3 Slope Sign Analysis (V)
   
   The preview here provides a context for slope or derivative calculations.
   

3.  Algebraic Preview Continued.
   
   4 Single Factor Analysis (V)
   4 Two Factor (V)
   4 More Factors (V)
   4 With Divisors (V)
   5 Max-Min Tests
   6 Discontinuities (optional)
   

4. Optional:  Solving Inequalities - Animated Examples.

Chapter I:  Preparation for Calculus:

1.  Arithmetic & Algebra Review  Exercises  to catch and correct common mistakes made by students entering calculus. 

   Problem Set	Answers
   A Arithmetic Problems	Solutions I
   B Caclulator Problems	Solutions II
   C More Arithmetic Problems	Solutions III
   D Problems with hints of Algebras	Solutions IV

   If you know your arithmetic skills are weak, watch a few Arithmetic WebVideos (Real Player Required). or visit the site areas on Solving Linear Equations with Stick Diagrams   and Fractions,  Ratios, Rates, Proportions   & Units  (Remember to budget your time. )
   

   If you do these problems, see the solutions to correct yourself, but also ask another who has passed calculus to correct your notation. That is a hassle. But imprecision or incorrect notation will cause you grief. 
   
    Markers for my assignments would be told to catch and correct all errors in notation and comprehension, so that my students have the chance to learn from their mistakes  
   

2. Logic Mastery - See the difference between one and two-way 2. Implication Rules.  Read Chains of Reason and Longer Chains of Reason. (mathematical induction). Logic mastery is a must for precision reading and writing in calculus.  Optional: 5 Knowledge Islands,  Imagine how a body of knowledge may have different entry points (or introductions).  But how entered will not affect the end result except for ease of travel. 
   

3. In calculus, we will talk about and describe numbers, amounts and quantities in a very precise and careful ways. Read the essay [What is a Variable]. and chapters 8 to 11. in the online book Three Skills for Algebra.  Pronoun Selection  or How to denote and name  Numbers and Quantity:  Chapter 12  in the online book Three Skills for Algebra 
   
   Pronoun: Is that the real thing or just a reference to it
   

4. Read now or when needed.  In Volume 2, Three Skills for Algebra, read  Mathematical Induction  then see the following chapters.[ 22. Geometric and Arithmetic Sums] [23 Summation Notation] [25 Mathematical Induction and Recursion Proofs, Product Notation, & Factorial Notation ] Meet the factorial function n!; summation notation and justification for the geometric summation formula. Summation notation will be needed in the discussion of integration and in the approximation of areas under  and between  curves y = f(x) 
   

5. Read Now or when Needed: 
   
   Distance Formula Etc
   Inequalities
   Solving Inequalities
   Function Domains Notes
   Domain Examples
   Polynomials - Domains  One Example, A Few More Might be warranted.
   Function Combinations  Addition, Subtraction, Product, Quotients
   
   Function Composition II  Ditto.
   Solving y**n = x**m

   Or what is the	m   n	root of x ?
   And what is xb when b =	m  n	?

Chapter 2: Limits, Numerically & Algebraically

The next lessons introduce the concepts of limits informally and then more carefully so that we can define derivatives and later other quantities via limits of approximations, and then see how to calculate derivatives or other quantities without mentioning limits. 

1. Real Numbers - Decimal View & Definition of Real Numbers. 
2. Limits Numerically. Examples are presented to introduce the limit concept.
3. Limit Properties. An algebraic description of the properties of limits.
4. What is a Limit. The definition codifies the concept. 
5. Decimal View Point of limits
6. Continuous Functions. This page answers the question, when can a limit be evaluated by immediate substitution.
7. Limits of Composite Functions - Needed for discussion of chain-rule, and implicit in evaluation of some limits. 
8. Limit Evaluations. See how immediate and delayed substitution can be used to evaluate limits.
9. Examples of Limit Evaluation - Animated Gifs.
10. Optional: Examples of One Sided Limit Examples - Animated Gifs
11. Optional: Examples of Limits involving Infinity - limits to infinity and limits that give infinity - Animated Gifs
12. Limits with Parameter. The limit process may eliminate dependence on a variable,  but leave dependence on another, the so called parameter. This concepts is needed 
13.  

Chapter 3.  Derivatives and Differentiation

Definition via limits and algebraic calculation.

1. Derivatives Motivation. We can not say exactly what the slope is for a nonlinear curve, but we can approximate it. If the limit of the approximation exist, we take that to be our codification and concept of the slope.
2. Derivatives More Motivation (Repetitious?)
   
3. Derivatives Defined -  Further discussion of the Limit based codification or mathematical definition of a derivative of a function a point. Here is a limit that depends on a parameter.
   
   The rest of this chapter covers the Sum, Product, Reciprocal, Quotient and Chain Rules first alone and then together to obtain methods for calculating more and more derivatives algebraically and quickly in place of direct limit evaluation. Limits were but a stepping stone for the clarification of the concept of what is the derivative or slope of a nonlinear function. 
4. Derivatives of Trig Functions.  See the geometric, decimal and algebraic reasoning leading to the derivatives of sines and cosines. On first reading assume the rules for differentiation of sines and cosines.  Leave the justification for later. 
   

   ---

5. Differentiation Methods Animated Examples  
   
   3 Derivatives of sin(x) & cos(x)
   3.Sum Rule - theory
   3. Product Rule 
   3. Power Rule
   3. Previous Rules Combined
   3. Polynomials, Derivatives of
   3. Reciprocal Rule
   3. Reciprocal Law (sec x  & csc x)
   3. Reciprocal Law  & Power Rule
   3. Power Law for Negative Integers
   3. Quotient Rule
   3. Quotient Rule Examples
   

   ---

6. Preparing for the Chain-Rule: 
   
   Chain Rule - Linear Functions . This discussion may be seen a digression or another prelude and context for the chain -rule. 
   
   Chain Rule Lite - Polynomials.  The chain rule where a power or  polynomial provides the outer function  provide an optional  intermediate step in the understanding of the chain rule when the outer function need not a polynomial.
   
   The statement and proof of the chain rule may be easier to digests after some  animated examples:  
   
   Chain Rule Examples I
   Chain Rule Examples II
   
7. Preparing for the Chain Rule - the key linear approximation.
   
8. Proof of the Chain Rule - General.  Here is a statement and a very simple rigourous, proof of the chain rule based on linear approximation, a proof that may help understanding.  The route  avoids the division by zero by exploiting linear approximation in place of quotient.  Derivatives of Inverse Fns are obtained as well from the linear approximation method. 
   

   ---

   More animated examples are being prepared (March 2006)  for addition after this point.

9. Chain Rule Examples with logarithms and exponentials, etc.
    Assumes the given differentiation formulas for logarithms and exponential functions.
   
10. More Differentiation Examples
11. Implicit Differentiation - to come with consequences: Power Rule Extended  to Rational Exponents - to come

Chapter 4: More on  Limits and Derivatives  

Consequences or Applications

1. Theory:  Necessary Conditions for maxima and minima follow  from Linear Approximation of functions. 
   
2. Revisit the Algebraic Preview Examples
   
   4 Single Factor Analysis (V)
   4 Two Factor (V)
   4 More Factors (V)
   4 With Divisors (V)
   
   and use your powers of differentiation to obtain the slope formula in these sign analysis problems. 
   
3. A Context for Second and Higher Derivatives:  See the online chapters 11 Slope of Slope & 13 Acceleration besides material elsewhere.  The first derivative of a function y  = f (x) is another function f'(x) which may (?) be differentiated again and again to get the second and higher order derivatives of the function f(x). Graphing of functions and the location of maxima and minima of the original function and its derivative depend on sign analysis of the first and second derivative.  To come: Examples Calculation and use of second derivatives, theory and practice.
   
4. The  Second Derivative Test when it applies, allows you classical solutions of f'(x) = 0 as local maxima or minima. 
5. Examples of how first and second derivatives, and asymptotes, etc may be used to sketch curves y = f(x)
   
   4. Sketch y = x^3 - 6x^2- 12x
   4. Sketch y = x^3 - 3 x^2 - 9x
   4. Skech y = 1 - 1/(1+x^2)
   
6. Volume 3, Chapter 16, What is Velocity - limits use to approximate and then codify (forma limit definition) concept of velocity.  The similar process was followed in the approximation and the codification (formal limit definition) of derivatives - the slope of y = f(x).

Chapter 5: Area as a Limit and Integration

1. Volume 3, Chapter 17, What is Area, - - this question  sets the stage for integration in the next chapter. Here again the limit of approximation is codified to provide the definition of area under a curve. 
   
2. Volume 3, Chapter 18, on Integration, or the First and Second Fundamental Theorems of Calculus.
   
   18 Integration -
   18 Area Calculation
   18 Function Definition in 6 Ways
   
   There are two fundamental theorems of calculus. The first deals with the existence of area - the convergence of approximations to the area. The second deals with the calculation of area via the reversal of the differentiation process: That is, given a function y  = f(x), find a function F(x) such that F'(x) = f(x).
   
3. Evaluation of Indefinite and Definite Integrals
   with the aid of anti-derivatives
   
   [5. Indefinite Integrals - First Examples] 
   [5. Indefinite Integrals - Chain Rule in Reverse]
   [5 Indefinite Integrals - More Examples]
   
   [6. Definite Integral - Evaluation Examples]
   [6. Area under y = x² + 3x from 1 to 3]
   
4. Exponentials and Logarithms: New Functions from old (i) by varying the upper limit of a definite integral, and (ii) by constructing the inverse function. One way to define a new function F(x) is to define it as the area under a curve t = f(r) from r = a to r  = x. This method leads to the  under-curve introduction of logarithms, their derivatives,  and the introduction of the exponential function as it inverse, and derivative of the exponential function.  Part of the story appears in the chapter 19 of Volume 3..
   
   19 Logs & Powers
   19 Natural Log.
   19 Exponential Functions.  
   
   Recall the Chain Rule - General. and the discussion at this site of the  Derivatives of Inverse Functions.
   
5. Logarithmic and Exponential Differentiation Rules.  Logarithmic Differentiation Method for products. - Derivation of Differentiation Rule for products with and without logs.
   
6. Methods of Integration - Differentiation Rules in Reverse,  Integration by Parts from Product Rule - Substitution Methods from use of Chain Rule in Reverse.
   
7. Approximation of arclength, areas and volumes via Riemann Sums (or differentials) and evaluation via integration. Volumes of Revolution. Quantities from Physics. ... ??

Extra Material: Theorems and Proofs for a first course in calculus.

First courses in calculus normally state theorems about limits, continuity and integration without proofs. The following pages provide the proofs.  

Triangle Inequality
Error Control Inequalities
Limit of a Sequence
Properties of Limits
Pigeon Hole Principle
Bolzano-Weierstrass Thm
Range & Continuity 
Equicontinuity 
Mean Value Theorem
Rolle's Theorem  
Constant Difference Thm
Lipshitz Continuity Thm
One Sided Range Thms

Intermediate Value Thms 
Riemann Sums I 
Riemann Sums II
Improper Integrals

The One Sided Range theorems  may be publishable in peer review journals which explore advances for the exposition of mathematics. They imply the image of an closed interval [a,b] under a continuous function f is a closed interval [c,d] = f([a,b]), and so imply or include the intermediate and extreme values theorems. The foregoing represents material for the gifted and talented in calculus or students who need to see proofs for the sake of their intellectual comfort. 

www.whyslopes.com
More Calculus
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Calculus Videos
0.   First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn:  Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
3. Derivative Motivation
3.  Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4.  Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5  Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere

To Learn More, visit Volumes 2 and 3.

Advanced Topics

Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity

To Learn More: Visit Real Analysis.

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