Posts Tagged ‘b’
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 »
