How do you find the taylor series for #f(x) = cos x # centered at a=pi?
The general formula for the Taylor Series is as follows:
Thus, we have to take the derivative multiple times. You should also consider the following:
Now, writing it out, we get:
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To find the Taylor series for ( f(x) = \cos x ) centered at ( a = \pi ), we can use the formula for the Taylor series expansion:
[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n ]
First, we find the derivatives of ( f(x) = \cos x ) up to the desired order at ( a = \pi ), which is the center of the Taylor series. Then, we substitute these derivatives into the Taylor series formula.
Here are the derivatives of ( \cos x ):
[ f(x) = \cos x ] [ f'(x) = -\sin x ] [ f''(x) = -\cos x ] [ f'''(x) = \sin x ] [ f^{(4)}(x) = \cos x ]
Now, evaluate these derivatives at ( x = \pi ):
[ f(\pi) = \cos \pi = -1 ] [ f'(\pi) = -\sin \pi = 0 ] [ f''(\pi) = -\cos \pi = -1 ] [ f'''(\pi) = \sin \pi = 0 ] [ f^{(4)}(\pi) = \cos \pi = -1 ]
Substitute these values into the Taylor series formula:
[ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} (x - \pi)^n ]
This is the Taylor series for ( \cos x ) centered at ( a = \pi ).
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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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