Posts Tagged ‘Calculus’
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
Posted in Calculus | No Comments »
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
Posted in Calculus | No Comments »
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
Posted in Calculus | No Comments »
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
Posted in Calculus | No Comments »
Thursday, November 5th, 2009
How to Use Parametrization
Description
A detailed tutorial on how to use parametrization. Step by step tutorial including several examples of how to use parametrization for reference.
Overview
Parametrization can be used in many different branches of math, including algebra and calculus. Parametrization involves setting up parameters necessary for the complete or relevent specification of a geometric object. This means it is only used when calculating a shape or part of a shape, because that is what a geometric object is. Sometimes, this is nothing more than identifying the parameters. Other times it becomes an involved mathematical process that is used to find out what the parameters are.
Tags: Calculus, complete, decide, deciding, define, defining, differential equations, geometric, identify, identifying, parameter, parametrization, relevent, set, setting, shape, specification, vector
Posted in Differential Equations | No Comments »
Friday, October 30th, 2009
How to Solve Work Rate Problems
Description
A detailed tutorial on solving work rate problems. Step by step tutorial including several examples of work rate problems for reference.
Overview
A work rate problem is a word problems that asks you to calculate the amount of time it will take to do something with two different rates of work. They first show up in basic algebra courses but work rate problems get more complicated and will continue on even in calculus. It is easier to solve work rate problems if you use a chart. First, you need to find the task rate – the rate at which each person is doing something. You do this by dividing the number of tasks (which should be one) by how many hours it takes them to finish it. Then you choose a variable for time. Your task will take that variable divided by the number of hours. You should come up with 2 (or more) results for task. Add these results together and have them equal the number of people there are total working on the task. Then solve for your time variable. Sometimes it will be difficult to solve for the time variable without using an algebra trick of multiplication to change the numbers a bit.
Tags: add, algebra, calculate, Calculus, chart, divide, hours, problem, proportion, rate, task, time, variable, word, work
Posted in Algebra | No Comments »
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
Posted in Calculus | No Comments »
Tuesday, October 27th, 2009
Introduction to Hilbert Space
Description
A detailed tutorial on on the application of Hilbert space. Step by step tutorial including several examples of Hilbert space for reference.
Overview
A Hilbert space is commonly used in vector algebra and calculus to generalize the notion of Euclidean space. It is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Hilbert spaces are also complete, which is a property that allows enough limits in the space for calculus to be used accurately.
Tags: abstract, algebra, angle, Calculus, complete, Hilbert, inner, length, limit, product, space, structure, vector
Posted in Algebra | No Comments »
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
Posted in Calculus | No Comments »
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
Posted in Calculus | No Comments »
