Math - Page 5

If you’re asked to find a derivative of some function, sometimes it’s not a trivial thing to do. If you’re lucky and the function is among common functions, you can write down the required derivative right away using table of derivatives if necessary. But what if given expression is more complicated? That’s what this section is about. Let’s discuss chain rule for derivatives or so called outside-inside rule. This rule is needed when we want to find derivative of non-common function which is in fact composition of two or more common functions. Continue reading →

Earlier we’ve talked about gamma and beta functions.You’re probably interested, what beta function is useful for? In this section we’ll show how to integrate certain functions in simpler way with the help of beta function. Suppose we need to evaluate the following integral:

\int_0^{3}{x^{1/2}(27-x^3)^{-1/2} dx}

At first glance we can say that there is not an easy expression to deal with under the integral sign. As you can see we have radicals and powers under the integral sign so it’s not an easy thing to integrate directly. Moreover, take a look at the expression in braces: as it’s in negative power it means this is a denominator; and at the upper limit this denominator turns into zero. In other words, we’re facing an improper integral. And that’s what Euler’s functions are about! But which one should  we take – gamma or beta?  Continue reading →

In the previous section we’ve discussed complex numbers, considered how the concept arose and have given some basic ideas about how to deal with this stuff. In this section we’re going to consider how to perform basic operations with complex numbers. This time we’ll go through every detail and show several examples. We’ll start with addition. Continue reading →

Do you know what is complex number? Or probably you’ve heard about imaginary unit? How did mathematicians happen to create them and what for? If you ever take a course of complex analysis you already know the answer. If you don’t, it’s not a problem either as getting the basics of complex numbers doesn’t require you to be a math master, school course is nearly enough. Historically the concept of complex numbers arose from the problem of solving quadratic equations. Continue reading →

Evaluation of integrals is sometimes a very complicated procedure, especially when we’re dealing with so called improper integrals. You can face this kind of tasks in integral calculus courses. Gamma and beta functions, or so called Euler integrals, allow to simplify evaluation of integrals for specially constructed functions. In fact, sometimes it’s possible to transform initial integral so that it’s reduced to calculation of gamma or beta function or both. Let’s consider an example. Continue reading →

One of common tasks in algebra homework is to solve a system of two linear equations. Along with other methods, we can do it graphically. This section is devoted to intersection of two straight lines on a plane. In other words, let’s talk about solving a system of two linear equations. As we discussed earlier, a straight line on a plane is described by the following equation:

y=kx+b

This equation sets a certain dependence between coordinates $x$ and $y$ of the point lying on this line. So, if we want to discuss intersection of two lines, we’ll have to deal with two equations of such kind.

Consider the following system:

\left\{ \begin{aligned}y=2x+3\\x-2y=0 \end{aligned}\right. Continue reading →

Let’s talk about straight lines on a plane. We often answer questions of our visitors about how to graph a line and other related stuff. For example, recently we were asked how to graph y=2x+3. To answer this question let’s consider equation of a straight line or so called linear equation on the plane. For that we will need a coordinate plane. It is formed by a horizontal number line called the x-axis and a vertical number line called the y-axis. The two axes intersect at a point called the origin. That’s not all. To define coordinates we should chose the scale Continue reading →