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Constructing Taylor series for Sine and Cosine

Taylor series is used for approximation of functions. Check our article about Taylor series if you need more details. As you know, often general formula is not enough for solving problems. There’s no need in reinventing the bicycle each time, when you face one of common functions. More profitable way is to memorize a few expansions and use ready formulas when required. Previously we’ve considered Taylor expansions for exponential and logarithm (click here fore details). Let’s proceed and find formulas for sine and cosine.

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Do Not Allow Your Homework to Make You Sick

Have you ever told your Mom like, ‘oh, I have a terrible headache, I can’t do my homework’? Did you really feel that way? Even if you sometimes lie about health problems to eschew doing your assignments, it doesn’t mean that you won’t have them in near future. It’s no longer a myth about students suffering physically while doing homework. But here we come to teach you what the major dangers are and how to manage studying process to reduce harm for your health. (more…)

How Planes Fly

Do you know how the planes fly? The physics of this process is the subject of our article. Of course, a plane is rather complex object and getting it in the air takes a bit of “rocket science” involved; thus, we should start with something simpler. Everybody knows how draft slaps doors. Sharp bangs happen when a fast flow of air goes through the door gap. This phenomenon is explained by Bernoulli’s principle, which is represented in liquids and gases. (more…)

How to Construct Taylor Series for Exponential and Logarithm

Recently we’ve discussed what Taylor series is (check our article for more information). In practice, it’s useful to know certain expansions of elementary functions. It’s like a table of derivatives or integrals. If you have it, you’re saving your time in calculations. The case of particular interest is the expansion of function in the vicinity of $x=0$ . Such series is called MacLaurin expansion, and that’s what we’re going to consider for the cases of some common functions. Maclaurin series of elementary functions is widely used and appears frequently in math tasks, so today we’ll obtain it for exponential and logarithmic functions. (more…)

How Important is Homework ?

What’s more important – to do your homework or to sleep? Probably every person said or thought “I hate homework” at least once. Even if you are a diligent student and strive to get all knowledge offered, doing almost every task and successfully completing all tests. This feeling of despair is inevitable and hardly avoidable at times of assignment overloads and especially during the finals week. The next question naturally arises – is all this homework business necessary, is it as important as teachers are trying to present? Practicing in class often seems quite enough, and the fact that most tasks are routine and thus boring adds fuel to the fire. Do we need homework?

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What is Taylor Series

Suppose you need to calculate $\sin{138}$ and don’t have a calculator at hand. How’d you do that? The way out is to approximate your function with something more convenient to work with, for example, polynomials: $x, x^2, x^3$ and so on. In this section, we’re going to discuss Taylor series which is an expansion of function into infinite sum of power functions. The series is called in honor of English mathematician Brook Taylor, though it was known before Taylor’s works. Taylor series is applied for approximation of function by polynomials. Such approach allows to replace initial more or less complicated function with the sum of simpler ones. Let’s get started. (more…)

Why Snow is White

Most often, the question “why is the snow white” is asked by children, yet not all adults know the answer as well. Why it’s white and not red, green or whatever? Moreover, there’s one more question concerning snow. Why is the ice transparent while the snow is white? The answer to both of these questions is connected with two things: the very notion of color and the structure of snow and ice. We’ll start with the first problem. Let’s explain it from physics perspective. (more…)