How do you find the taylor series for #f(x)=xcos(x^2)#?

Answer 1
The quickest way is to first recall the Taylor series for cosine about #x=0#:
#cos(x)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+\cdots#
Now just replace all these #x#'s by #x^2#'s and then multiply everything by #x#:
#xcos(x^2)=x-x^5/(5!)+x^9/(9!)-x^13/(6!)+\cdots#.
This can be shown to converge for all #x#.

You could also use the formula

#f(0)+f'(0)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)x^3+\cdots#, but that would be a real big pain for this problem.
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Answer 2

To find the Taylor series for ( f(x) = x \cos(x^2) ), you need to express ( f(x) ) as a sum of terms involving powers of ( x ). The Taylor series expansion of ( f(x) ) around a point ( a ) is given by:

[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n ]

First, find the derivatives of ( f(x) = x \cos(x^2) ) up to the desired order. Then, evaluate these derivatives at the point ( a ), and substitute them into the Taylor series formula.

Let's start by finding the derivatives of ( f(x) ):

[ f(x) = x \cos(x^2) ]

[ f'(x) = \cos(x^2) - 2x^2 \sin(x^2) ]

[ f''(x) = -4x \sin(x^2) - 4x^3 \cos(x^2) ]

[ f'''(x) = -4\sin(x^2) - 12x^2 \cos(x^2) - 12x^4 \sin(x^2) ]

Now, we need to evaluate these derivatives at a specific point ( a ). For simplicity, let's choose ( a = 0 ).

[ f(0) = 0 ] [ f'(0) = 1 ] [ f''(0) = 0 ] [ f'''(0) = -4 ]

Now, substitute these values into the Taylor series formula:

[ f(x) \approx 0 + 1 \cdot x + 0 \cdot \frac{x^2}{2!} - 4 \cdot \frac{x^3}{3!} ]

Simplify the terms:

[ f(x) \approx x - \frac{4x^3}{3!} ]

This is the Taylor series for ( f(x) = x \cos(x^2) ) centered at ( a = 0 ), which is also known as the Maclaurin series.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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