How can you find the taylor expansion of #f(x) =sinx# about x=pi/6?
So we have:
and clearly in general:
The Taylor series is then:
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To find the Taylor expansion of ( f(x) = \sin(x) ) about ( x = \frac{\pi}{6} ), follow these steps:
 Determine the derivatives of ( f(x) = \sin(x) ) up to the desired order.
 Evaluate these derivatives at ( x = \frac{\pi}{6} ).
 Write the Taylor expansion using the derivatives and their respective coefficients.
Here are the detailed steps:

Derivatives of ( f(x) = \sin(x) ):
 First derivative: ( f'(x) = \cos(x) )
 Second derivative: ( f''(x) = \sin(x) )
 Third derivative: ( f'''(x) = \cos(x) )
 Fourth derivative: ( f^{(4)}(x) = \sin(x) )
 Fifth derivative: ( f^{(5)}(x) = \cos(x) )
 Sixth derivative: ( f^{(6)}(x) = \sin(x) )
 Seventh derivative: ( f^{(7)}(x) = \cos(x) )
 Eighth derivative: ( f^{(8)}(x) = \sin(x) )
 Continuing in this pattern, the derivatives alternate between sine and cosine functions.

Evaluate the derivatives at ( x = \frac{\pi}{6} ):
 ( f\left(\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} )
 ( f'\left(\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} )
 ( f''\left(\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} )
 ( f'''\left(\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} )

Write the Taylor expansion using the derivatives and their respective coefficients:
 ( \sin(x) = f\left(\frac{\pi}{6}\right) + f'\left(\frac{\pi}{6}\right)(x  \frac{\pi}{6}) + \frac{f''\left(\frac{\pi}{6}\right)}{2!}(x  \frac{\pi}{6})^2 + \frac{f'''\left(\frac{\pi}{6}\right)}{3!}(x  \frac{\pi}{6})^3 + \cdots )
Substitute the evaluated derivatives into the expansion to get the Taylor series expansion of ( \sin(x) ) about ( x = \frac{\pi}{6} ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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