Archive for the ‘Calculus’ Category
Thursday, November 19th, 2009
How to Find the Common Ratio of a Geometric Series
Description
A detailed tutorial on how to find the common ratio of a geometric series. Step by step tutorial including several examples of the common ratio for reference.
Overview
The common ratio is part of a geometric series, used commonly in calculus. The common ratio is the ratio of each term to the next – in other words, the common ratio is the pattern that the series or sequence follows. This is possible because in a geometric series, terms are only being multiplied by one number to get the next number, and it is always the same number. If a series is not geometric, it will not have a common ratio.
Tags: Calculus, common, geometric, multiplication, multiply, number, pattern, ratio, sequence, series, term
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Thursday, November 5th, 2009
Saddle-Point Approximation Explained
Description
A detailed tutorial on saddle-point approximation. Step by step tutorial including several examples of saddle-point approximation for reference.
Overview
Saddle-point approximation is also referred to as the method of steepest descent and Laplace’s method. It is a way of approximating integrals in the form 
. f(x) is some twice-differentiable function, M is a large number, and the integral endpoints a and b have a possibilty of being infinite.
Tags: a, approximation, b, Calculus, descent, differentiable, function, infinite, infinity, integral, Laplace, large, m, method, number, point, saddle, saddle-point, steepest, twice, twice-differentiable
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Thursday, November 5th, 2009
Overview of the Dominated Convergence Theorem
Description
A detailed tutorial on the dominated convergence theorem. Step by step tutorial including several examples of the dominated convergence theorem for reference.
Overview
Unlike the monotone convergence theorem, the dominated convergence theorem only has one form. The official name of the theorem is Lebesgue’s Dominated Convergence Theorem, but most people just call it the dominated convergence theorem. It is considered to be a special version of the Fatou-Lebesque theorem, so Fatou’s lemma is used in direct proofs of this theorem. This theorem is also closely related to the bounded convergence theorem.
Tags: bounded proof, Calculus, convergence, direct, dominated, Fatou, form, Lebesque, lemma, monotone, special, theorem, version
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Thursday, November 5th, 2009
Overview of the Monotone Convergence Theorem
Description
A detailed tutorial on the monotone convergence theorem. Step by step tutorial including several examples of the monotone convergence theorem for reference.
Overview
There are several different theorems that the term “monotone convergence” can apply to. However, the most important one, and the one most common called the monotone convergence theorem, is the Lebesgue Monotone Convergence Theorem. This particular monotone convergence theorem deals with calculus, and with integrals and limits specifically. It is a more general form of the other two monotone convergence theorems, which is why it is considered to be the most important.
Tags: Calculus, converge, convergence, form, general, integral, Lebesgue, limit, monotone, number, real, sequence, series, theorem
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Friday, October 30th, 2009
How to Find Higher Order Derivatives
Description
A detailed tutorial on higher order derivatives. Step by step tutorial including several examples of higher order derivatives for reference.
Overview
A higher order derivative is a derivative with a power other than one – that is, a derivative is referred to as a first derivative, and the higher order derivatives are a second derivative, third derivative, etc. The second derivative is the derivative of the first derivative, and the third derivative is the derivative of the second derivative. When you know all the rules of taking derivatives, taking second and third derivatives are simple. Simply take the derivative and pretend it is another equation. When you go up beyond the third derivative this can get more challenging, as there will be many more parts to the equation.
Tags: antiderivative, Calculus, chain, derivative, First, higher, integral, order, power, product, quotient, rule, second, third
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Thursday, October 22nd, 2009
How to Identify a Concave Function
Description
A detailed tutorial on concave functions. Step by step tutorial including several examples of concave functions and concave down curves for reference.
Overview
When a function forms the graph of a curve, there are two types of functions it could be: a convex function, or a concave function. In this tutorial, we will discuss concave functions. A concave function is one with the endpoints facing down, forming the shape of an upside down bowl. When looking at the graph of a concave function, we say that it is concave down. Concavity can be found by the second derivative test in calculus.
Tags: Calculus, concave, concavity, convex, curve, derivative, down, endpoint, equation, function, graph, interval, second, test, up
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Thursday, October 22nd, 2009
How to Identify a Convex Function
Description
A detailed tutorial on convex functions. Step by step tutorial including several examples of convex functions and concave up curves for reference.
Overview
When a function forms the graph of a curve, there are two types of functions it could be: a convex function, or a concave function. In this tutorial, we will discuss convex functions. A convex function is one with the endpoints facing up, forming the shape of a bowl. When looking at the graph of a convex function, we say that it is concave up. Concavity can be found by the second derivative test in calculus.
Tags: Calculus, concave, concavity, convex, curve, derivative, down, endpoint, equation, function, graph, interval, second, test, up
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Friday, October 16th, 2009
Overview of the Conjugate Zeros Theorem
Description
A detailed tutorial on the conjugate zeros theorem. Step by step tutorial including several examples of the conjugate zeros theorem for reference.
Overview
The conjugate zeros theorem states that if a + b * i is a zero of a polynomial with real coefficients, then so is a – b * i. The conjugate zeros theorem can be proved by taking any function in this form and setting it equal to zero. The conjugate zeros theorem makes many equations easier to solve, especially complex equations when you get to higher levels of math.
Tags: a, b, Calculus, complex, conjugate, equations, function, i, imaginary, Math, number, theorem, zero, zeros
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Friday, October 9th, 2009
Witch of Agnesi Explained
Description
A detailed tutorial of the Witch of Agnesi. Step by step tutorial including a visual example of the Witch of Agnesi for reference.
Overview
The Witch of Agnesi is actually a curve. This curve can be a circle, or it can be a regular curve. The movement of the curve flows up and down, and the curve itself changes as it moves. This curve is defined by the Cartesian equation 
.
It is called the Witch of Agnesi by a simple mistranslation into English. This curve was named in Italian – la versiera di Agnesi, which means the Curve of Agnesi. When translating the name, “la versiera” was accidentally read as “l’awersiera”, which means a woman who is contrary to God, or a demon or witch. Hence it was called the Witch of Agnesi.
Tags: Calculus, cartesian, circle, curve, equation, l'awersiera di Agnesi, la versiera di Agnesi, Maria Agnesi, Math, Witch of Agnesi, Witch of Maria Agnesi
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Friday, October 9th, 2009
Lissajous Curve Explained
Description
A detailed tutorial of a lissajous curve. Step by step tutorial including several visual examples of lissajous curves for reference.
Overview
A Lissajous curve represents the graph of a system of parametric equations, which can be mathematically expressed as 
. This also decribes complex harmonic motion. The way that the figure appears is very sensitive to the ratio a / b, so the figure can appear in many different forms.
Tags: Bowditch, Calculus, complex, curve, equation, figure, form, graph, harmonic, Lissajous, Math, motion, paramentric, ratio, system
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