Posts Tagged ‘a’
Tuesday, October 6th, 2009
Overview of Existential and Universal Quantification
Description
A detailed tutoral on existential and universal quantification. Step by step tutorial including several examples of existential and universal quantification for reference.
Overview
Existential and universal quantifiers give us different ways to write expressions and mathematical equations. The existential quanitfier looks like a backwards capital E, and basically means “some”. The universal quantifier looks like an upside down A, and basically means “all”. For example, take the sentence “Some children don’t like clowns.” In the mathematical form of quantifiers, this would be written as (Ex) (x is a child) ^ (Ay) (y is a clown) –> (x does not like y). “Some children” indicates that you would use an existential quantifier, not a universal quantifier. Since clowns in not specific, based on context we must assume that the statement refers to all clown, and therefore we use the universal quantifier. The ^ is the symbol for “and”, implying that both of these statements are true, and the arrow is an implication stating that if there is a clown, some children will not like it based on the previous statement.
Tags: a, all, discrete math, E, existential, logic, Math, quantification, quantifiers, some, statement, universal
Posted in Discrete Math | No Comments »
Tuesday, October 6th, 2009
How to Test for Convergence Using the Geometric Series Test
Description
A detailed tutorial on how to test for convergence using the geometric series test. Step by step tutorial including several examples of testing for convergence using the geometric series test for reference.
Overview
A geometric series is a series that maintains a constant ratio between a set of terms. This series is an addition series, and would be expressed as 1/a + 1/2a + 1/4a, extending as far as you wish in either direction. If a series does not have that constant ratio, then it is not a geometric series. The series should converge at one, because as all the numbers are added they get closer and closer to one. The first term of a geometric series is given by a, and the ratio of a geometric series is given by r. If the ratio is less than one, then the geometric series converges to a / (1 – r). If the ratio is greater than or equal to one, then the series diverges. Usually the series will converge, which is why this is considered a test for convergence and not for divergence.
Tags: a, addition, Calculus, converge, convergence, diverge, divergence, equal to, first term, geometric, greater than, less than, Math, notation, r, ratio, series, summation, test
Posted in Calculus | No Comments »
Friday, October 2nd, 2009
The Union and Intersection of Sets
Description
A detailed tutorial on the union and intersection of sets. Step by step tutorial including several examples of the union and intersection of sets for reference.
Overview
Set theory is a branch of mathematics that deals with sets of numbers and the way that they interact with each other. One part of set theory is union and intersection. Union is represented by the symbol U, and means to combine the numbers in a set. The union of A and B states that for all x in A and B, the union contains all x in A and all x in B. Intersection is represented by an upside-down letter U, and means to only use numbers that are found in both sets. The intersection of A and B states that for all x in A and B, the intersection contains all x found in both A and B. The definitions might seem similar, but they are different.
Union:
A = {1, 2, 3, 4}, B = {2, 3, 6, 7}. The union would be {1, 2, 3, 4, 6, 7}.
Intersection:
A = {1, 2, 3, 4}, B = {2, 3, 6, 7}. The intersection would be {2, 3}
Tags: a, and, b, belonging to, combine, difference, discrete math, interact, intersection, Math, or, set theory, sets, union, x
Posted in Discrete Math | No Comments »
Thursday, October 1st, 2009
Introduction to Fermat’s Last Theorem
Description
A detailed tutorial of Fermat’s Last Theorem. Step by step tutorial including several examples of Fermat’s Last Theorem for reference.
Overview
Fermat’s Last Theorem is one of the most well known mathematical theorems. Fermat’s Last Theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. Notice that the pattern for this theorem follows the Pythagorean theorem. This theorem had to be proved for odd prime numbers, as Fermat had only left that there was the special instance of n = 4 that works for this equation. Fermat first came up with the problem in 1637, but it was not solved until 1995. This theorem led to the developement of both algebraic number theory and the proof of the modularity theorem.
Tags: a, algebraic number theory, Andrew Wiles, b, c, Calculus, Fermat's Last Theorem, integers, Math, modularity theorem, n!, numbers, odd, Pierre de Fermat, positive, prime, pythagorean theorem
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Boundedness Theorem Explained
Description
A detailed tutorial of the boundedness theorem. Step by step tutorial including an explanation of the boundedness theorem for reference. Knowledge of the boundedness theorem is required in calculus.
Overview
The boundedness theorem is a theorem that is very closely linked to the extreme value theorem. The boundedness theorem states that a continuous function f in the closed interval [a, b] is bounded on that interval. In mathematical terms, this means that there exist real numbers m and M such that ![m \le f(x) \le M\quad\text{for all }x \in [a,b].\,](http://s.wordpress.com/latex.php?latex=m%20%5Cle%20f%28x%29%20%5Cle%20M%5Cquad%5Ctext%7Bfor%20all%20%7Dx%20%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "m \le f(x) \le M\quad\text{for all }x \in [a,b].\,")
This translates to mean “m is less than or equal to f(x) which is less than or equal to M for all x belonging to [a, b]“.
Tags: a, b, bounded, boundedness theorem, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, m, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Thursday, October 1st, 2009
Extreme Value Theorem Explained
Description
A detailed tutorial of the extreme value theorem. Step by step tutorial including an explanation of the extreme value theorem for reference. Knowledge of the extreme value theorem is required in calculus.
Overview
The extreme value theorem states that if a real valued function f is continuous in the closed and bounded interval [a, b], then f must attain its maximum and minimum value at least once. In mathematical terms, this means that there exist numbers c and d in [a, b] such that ![f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,](http://s.wordpress.com/latex.php?latex=f%28c%29%20%5Cge%20f%28x%29%20%5Cge%20f%28d%29%5Cquad%5Ctext%7Bfor%20all%20%7Dx%5Cin%20%5Ba%2Cb%5D.%5C%2C&bg=ffffff&fg=000000&s=0 "f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b].\,")
The translation of that formula is “f(c) is greater than or equal to f(x) which is greater than or equal to f(d), for all x belonging to [a, b]“. In order for something to belong to an interval, it must be found in the interval.
Tags: a, b, bounded, c, Calculus, closed, continuous, d, EVT, extreme value theorem, f(c), f(d), f(x), function, graph, local, Math, maxima, maximum, minima, minumum, value, x
Posted in Calculus | No Comments »
Thursday, September 24th, 2009
Intermediate Value Theorem Explained
Description
A detailed tutorial of the intermediate value theorem. Step by step tutorial including an explanation of the intermediate value theorem for reference. Knowledge of the intermediate value theorem is required in calculus.
Overview
The intermediate value theorem states that for each value between the upper bound and the greatest lower bound of the graph of a continuous function that there is a corresponding value in its domain. In mathematical terms, the intermediate value theorem states that if f is a continuous function on the closed interval [a, b] and M is a number between f(a) and f(b), then there exists at least one number c that f(c) = M. When writing proofs in calculus, you can say that something has been proven by the IVT if you used the intermediate value theorem to reach your conclusion.
Tags: a, b, c, Calculus, continuous, corresponding, domain, f(a), f(b), f(c), function, graph, greatest lower bound, intermediate value theorem, IVT, m, Math, upper bound, value
Posted in Calculus | No Comments »
Thursday, September 17th, 2009
Definition of the Mean Value Theorem
Description
A detailed tutorial on the solving of the Mean Value Theorem. Step by step tutorial including several examples of how to solve the Mean Value Theorem for reference.
Overview
You can easily figure out what the Mean Value Theorem is by looking at the word mean – a mean is an average. The Mean Value Theorem states that there is at least one point on the graph of a function where the derivative is equal to the average slope of the entire section of the graph you are looking at. The requirements are that the graph is both continuous and differentiable on the interval [a, b], where a < b. Then there exists some c in (a, b) such that:
f ‘(c) = [f(b) - f(a)] / [b - a]
The Mean Value Theorem is very similar to Rolle’s Theorem, which is a more specific theorem stating the same thing.
Tags: a, average, b, c, Calculus, continuous, derivative, differentiable, interval, Math, mean, mean value theorem, rolle's theorem, slope, theorem, value
Posted in Calculus | No Comments »
