Taylor Series of cos(3x^2)

In summary, the conversation discusses determining the Taylor series for the function cos(3x2) at x = 0 by computing P5(x), which requires the Maclaurin series for degree 5. The attempt at a solution involves using a shortcut or heuristic, but it is ultimately solved by stopping at the fourth degree, resulting in the answer of 1 - 9x4/2.
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Kaura
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Homework Statement


Determine the Taylor series for the function below at x = 0 by computing P5(x)
f(x) = cos(3x2)

Homework Equations


Maclaurin Series for degree 5

f(0) + f1(0)x + f2(0)x2/2! + f3(0)x3/3! + f4(0)x4/4! + f5(0)x5/5!

The Attempt at a Solution


I know how to do this but attempting to solve the 3rd derivative of cos(3x2) and onward is simply infeasible due to it requiring multiplication rule and stuff
I remember my professor mentioning some sort of short cut to certain series
Is there a short cut or heuristic to solve this or do I simply have to solve the higher order derivatives?

Update
I tried solving the series as cos(u) where u = 3x2 and got
1 - 9x4/2 + 27x8/8
which matches the result from a Taylor Series calculator online
I feel like I am making a basic mistake right now please enlighten me

Update
Genius me did not realize that I needed to stop at the 4th degree even after doing to replacement
1 - 9x4/2
was accepted as the correct answer
I guess I ended up answering my own question
 
Last edited:
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  • #2
Yes, you did. And a good job too.
 

1. What is a Taylor Series?

A Taylor Series is a representation of a function as an infinite sum of terms, where each term is a polynomial function of the variable x.

2. How is the Taylor Series of cos(3x^2) calculated?

The Taylor Series of cos(3x^2) can be calculated using the Maclaurin Series, which is a special case of the Taylor Series where the expansion is centered at x=0. The formula for the Maclaurin Series of cos(3x^2) is:
cos(3x^2) = 1 - (9x^4)/2! + (81x^8)/4! - (729x^12)/6! + ...

3. What is the significance of the Taylor Series for cos(3x^2) in mathematics?

The Taylor Series of cos(3x^2) is significant because it allows us to approximate the value of cos(3x^2) at any point in its domain, using a finite number of terms in the series. This is useful in solving problems in physics, engineering, and other fields that involve trigonometric functions.

4. Can the Taylor Series of cos(3x^2) be used to find derivatives?

Yes, the Taylor Series of cos(3x^2) can be used to find derivatives of cos(3x^2). Differentiating the series term by term, we can obtain the derivatives of cos(3x^2) at any point in its domain.

5. How accurate is the Taylor Series of cos(3x^2)?

The accuracy of the Taylor Series of cos(3x^2) depends on the number of terms used in the series. The more terms we include, the more accurate our approximation of cos(3x^2) will be. However, because it is an infinite series, the exact value of cos(3x^2) can never be obtained, only approximated.

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