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YOU are better than YOU think. Show yourself how:
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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
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Caution: Site advice is approximately
correct, for some circumstances, not all. . That leaves room for thought and
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This WiZiQ
Calculus Tutorial Link gives the date and time of a
whyslopes. com interactive session in which students may see written and
spoken answers to start of calculus questions on say on functions,
limits, continuity, and derivatives. First session is free.
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
-
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.
-
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.
-
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)
-
Optional: Solving
Inequalities - Animated Examples.
-
Arithmetic & Algebra Review
Exercises to catch and correct common mistakes made by
students entering calculus.
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
-
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.
-
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
-
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)
-
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
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Or what is the |
m
n |
root of x ? |
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And what is xb when b = |
m
n |
? |
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A Digression:
Limits and Derivatives -Theory and Practice.
In the geometric and algebraic previews, the slopes of
curve y = f(x) is assumed to exist and formulas are given for it, so that
the shape of the curve or hill y = f(x) can be determined
qualitatively if not precisely from a sign analysis of formulas for
slopes.. That provides a context and initial reason for the calculation of
derivatives.
In calculus, we are concerned not only with calculations but also with
the justification or origins of formulas. The derivative of a
function at a point is a human invention (at least on planet earth).
Diagrams may suggest approximations to what the derivative at a point on a
curve should be. Saying how to compute a number, here the
derivative, directly or via a limit defines it. the derivative via a limit
or sequence of approximations is a human convention in calculus. The
limit, it it exists, of those approximations gives a definition the
derivative (or slope) at the point. Those limit calculation appear in the
first instance to be far from the formulas used in sign analysis, but in
an extension of your algebraic thinking skills, you will hopefully learn
how formulas for derivatives (slopes) come from formulas for functions via
rules which can be applied without mentioning limits.
Common Calculus Theme: Define and calculate a quantity using
limits, then calculate via algebraic rules without mentioning the
underlying limits.
This works for derivatives, integration and physical
quantities that can be formally defined via limits, limits that give
derivatives or integrals. Details to explain what is meant follow. Bon
Appetit.
|
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.
- Real Numbers - Decimal View &
Definition of Real Numbers.
- Limits Numerically. Examples are
presented to introduce the limit concept.
- Limit Properties. An algebraic
description of the properties of limits.
- What is a Limit. The
definition codifies the concept.
- Decimal View Point of limits
- Continuous Functions. This
page answers the question, when can a limit be evaluated by immediate
substitution.
- Limits of Composite
Functions - Needed for discussion of chain-rule, and implicit in
evaluation of some limits.
- Limit Evaluations.
See how immediate and delayed substitution can be used to evaluate limits.
- Examples of Limit Evaluation -
Animated Gifs.
- Optional: Examples of One Sided
Limit Examples - Animated Gifs
- Optional: Examples of Limits involving
Infinity - limits to infinity and limits that give infinity - Animated
Gifs
- 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
-
Definition via limits and algebraic calculation.
- 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.
- Derivatives More Motivation
(Repetitious?)
- 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.
- 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.
- 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
- 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
- Preparing for the Chain Rule - the
key linear approximation.
- 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.
- Chain Rule
Examples with logarithms and exponentials, etc.
Assumes the given differentiation formulas for logarithms and
exponential functions.
- More Differentiation Examples
- Implicit Differentiation - to come with consequences: Power Rule
Extended to Rational Exponents - to come
Consequences or Applications
- Theory: Necessary Conditions for maxima and minima
follow from Linear Approximation
of functions.
- 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.
- 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.
- The
Second Derivative Test when it applies, allows you classical solutions
of f'(x) = 0 as local maxima or minima.
- 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)
- 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).
- 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.
- 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).
- 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]
- 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.
- Logarithmic and Exponential Differentiation Rules.
Logarithmic Differentiation Method for products. - Derivation of
Differentiation Rule for products with and without logs.
- Methods of Integration - Differentiation Rules in
Reverse, Integration by Parts from Product Rule - Substitution Methods
from use of Chain Rule in Reverse.
- Approximation of arclength, areas and volumes via Riemann
Sums (or differentials) and evaluation via integration. Volumes of
Revolution. Quantities from Physics. ... ??
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.
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www.whyslopes.com
More Calculus
[ Next ]
For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
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.
[ Next ]
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