Which is the best of the following polynomial approximations to cos 2x near x=0?

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Please explain how to do the problem!

The Maclaurin Series for f(x) = cos x is 1 – x^2/2! + x^4/4! – …
Replacing x with 2x, we get
cos(2x) = 1 – (2x)^2/2! + (2x)^4/4! – … = 1 – 2x^2 + 2x^4/3 – …
So, the answer is e).
I hope this helps!

What our team says

Which is the best of the following polynomial approximations to cos 2x near x=0?

In mathematics, a polynomial is an expression consisting of variables and coefficients, that is, terms of the form ax^n. Polynomials are used to approximate other functions, in particular, trigonometric functions such as cosine. In this article, we will be looking at the best of the following polynomial approximations to cos 2x near x=0.

Taylor’s polynomial

1. Taylor’s polynomial is a good approximation to cos x near x=0. It is a polynomial of degree n that is equal to the sum of the first n terms of the Taylor series for cos x.

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2. Lagrange’s polynomial is a good approximation to cos x near x=0.5. It is a polynomial of degree n that is equal to the sum of the first n terms of the Lagrange interpolating polynomial for cos x.

3. Newton’s divided difference interpolating polynomial is a good approximation to cos x near x=1. It is a polynomial of degree n that is equal to the sum of the first n terms of Newton’s divided difference interpolating polynomial for cos x.

Pade’s polynomial

Pade’s polynomial is a good approximation to cos x near x=0.

Pade’s polynomial is a rational function that approximates cos x near x=0. It is of the form f(x)=ax^2+bx+c, where a, b, and c are constants. Pade’s polynomial has the advantage of being easy to compute and requiring only a few terms to achieve good accuracy.

Another advantage of Pade’s polynomial is that it can be used to approximate other functions besides cos x. For example, it can be used to approximate the function f(x)=1/(1+x^2) near x=0. This makes Pade’s polynomial a very versatile approximation tool.

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Overall, Pade’s polynomial is a good choice for approximating cos x near x=0. It is easy to compute and requires only a few terms to achieve good accuracy. Additionally, it can be used to approximate other functions besides cos x.

Chebyshev’s polynomial

Chebyshev’s polynomial is the best of the following polynomial approximations to cos x near x=0.

Chebyshev’s polynomial has several advantages over other polynomials. First, it converges faster than other polynomials. Second, it is more accurate than other polynomials. Third, it is easier to compute than other polynomials.

Chebyshev’s polynomial is the best choice for approximating cos x near x=0.

Conclusion

In this article, we have explored the different polynomial approximations to cos 2x near x=0. We have seen that each approximation has its own merits and drawbacks, and it is up to the individual to decide which one is best for their needs. In general, however, we can say that the Taylor series expansion is the most accurate of the three methods near x=0.

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