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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.
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in reading and
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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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Domains of Functions
The domain of a function is the set of values for which it is defined.
Sometimes it is easier to say where the function is not defined.
We assume our functions are real valued.
So values of the independent variable which would involve
the square root of negative numbers and hence complex numbers for values are
not allowed. Follow that convention until you are in course or a topic
involving complex-valued functions.
Domains of Basic Functions
- Polynomials p(u) are defined for all real numbers
- Rational functions (the ratio p(u)/g(u) of two polynomials) are
defined at points where division by zero appears, that is where g(u) =
0.
- sin(u) and cos(u) are define for all real numbers u. .
- The ratios and reciprocals of the sin(u) and cos(u) are defined except
those points where denominators would equal zero.
- logarithms are defined only for non-negative real numbers - when functions
are restricted to being real-valued.
- square roots and even roots are defined only for non-negative real numbers
- when functions are restricted to being real-valued.
- odd roots are defined for all real numbers.
- Powers xq where q = m/n and m = 2k is even and n odd is define
for all real numbers. Here xq = (xm)1/n
= (x1/n)m . Moreover, xq
= sign(x) |x|q
- Powers xq where q = m/n and m is odd even and n even is
define for all non-negative numbers. Here xq = (xm)1/n
= (x1/n)m But
both (xm)1/n and (x1/n)m
requires the computation of an even root of a negative number when x
is negative and n is even. So is not possible for real numbers when
functions are not allowed to be complex-valued.
- When q = m/n and a > 0, aq = exp( q ln(a)).
- When q = m/n and n is odd, and x is non-zero, |x|q
= exp( q ln(|x|)) (Why?) and xq = sign(x)|x|q
= sign(x)*exp(q*ln(|x|))
Composition of these basic functions yield further functions. The
domain of the resulting composite function depends on the domains of the
composed functions. Basic and composite functions may then appear in sums,
differences, products and quotients to provide further further functions.
Reciprocals and Quotients of functions, basic or composite or more
complicated, introduces points where the further function is not defined
in order to avoid division by zerio.
Remark: A finer discussion of the connections between powers,
logarithms and exponentials is possible. In a second course on calculus,
one can show that that the properties of natural logarithms and exponential
function imply f(x) = exp(x ln(a)) is a continuous function with the
property that f(q) = aq when a > 0 and q is a rational
number, positive, negative or zero. So we define ax = exp(x ln(a))
for all real numbers x.
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More Calculus
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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.
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