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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.
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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.
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Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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Decimal Insights on Limits, Continuity, Convergence
Decimal and decimal-free error-control perspectives of continuity, limits and
Cauchy sequences are given below. These perspectives is followed by comments on
math education.
Continuity and Unlimited Error Control
Limits and continuity in calculus may be described geometrically, that is,
intuitively and informally, or more precisely in terms of say epsilons and
deltas. The roles of epsilon and delta below are played by E > 0 and D >
0.
Imagine for instance we want to compute a function f(x) at the point x = A
accurately. So we can ask the error control question how close must x be to A in
order for f(x) to agree with f(A) to say k-decimal places. The answer might be
that x must agree with A to m decimal places. In some computational problems,
this answer for a specified number k of decimal places may be all that is
needed. But in other situations, we want in practice or in principle, unlimited
error control. Here we may want to say for any k, there is an m such f(x) will
agree with f(A) to k decimals if x agrees with A to m decimals. Unlimited error
control offers motivation and a perspective on the discussion of continuity.
Now will say that f(x) is continuous at x = A if for each whole number
k, there is a number m such the limit f(a) and the value of f(x) will agree to
k-decimals whenever the number x agrees with the value of a to m decimal
places. Continuity here represents the concept of unlimited error control in
decimal computations.
More generally, we can ask (following Cauchy), given an error control target
E > 0, how close must x be to A for the difference of f(x) and f(A) to be
less than E in magnitude? The answer follows by obtaining a number D with the
property that if |x-A| < d then |f(x) - f(A)| < E.
Without reference to decimals we can say that f(x) is continuous at x = A if
for every error control tolerance E > 0, there is a number D > 0 such that
whenever |x-A| < d then |f(x)- f(A)| < E. Here continuity at
x="A" corresponds to the idea of unlimited error control at
x="A." This second concept is decimal free. It is traditional to use
epsilons and deltas in place of E> 0 and D > 0.
Limits
We will say that a number L is the limit of a function f(x) as x approaches A
if one of the following conditions hold:
- (Decimal Perspective): For every whole
number k, there exist a whole number m such f(x) will agree will L to k
decimal places if x agrees with A to m decimal places.
- (Decimal Free Perspective): For every
positive number E > 0, there exists a positive number D> 0 such that
if |x-A| < d then |f(x)- f(A)| < E.
Both conditions are equivalent. Each implies the other. Why or how depends on
how you think of (or represent) the real numbers. For most people, assuming that
real numbers are represented by signed decimal expansions (infinite or finite)
is sufficient. Modern mathematics has alternate decimal (or base) -free
representations of real numbers.
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www.whyslopes.com
More Calculus
[ Back ] [ Up ] [ Next ]
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.
[ Back ] [ Up ] [ Next ]
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