How do you find the taylor series series for #f(x) = x^4  x^2 + 1# at c=1?
There are two methods:
The binomial theorem (Pascal's triangle ) is helpful in doing the expansion:
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To find the Taylor series for ( f(x) = x^4  x^2 + 1 ) at ( c = 1 ), we'll need to find the derivatives of ( f(x) ) at ( c = 1 ) and evaluate them, and then use these values to construct the Taylor series. Here are the steps:

Find the derivatives of ( f(x) ):
[ \begin{align*} f(x) & = x^4  x^2 + 1 \ f'(x) & = 4x^3  2x \ f''(x) & = 12x^2  2 \ f'''(x) & = 24x \ f''''(x) & = 24 \end{align*} ]

Evaluate the derivatives at ( c = 1 ):
[ \begin{align*} f(1) & = (1)^4  (1)^2 + 1 = 1 + 1 + 1 = 3 \ f'(1) & = 4(1)^3  2(1) = 4  (2) = 2 \ f''(1) & = 12(1)^2  2 = 12  2 = 10 \ f'''(1) & = 24(1) = 24 \ f''''(1) & = 24 \end{align*} ]

Construct the Taylor series using these derivatives:
[ \begin{align*} f(x) & = f(1) + f'(1)(x + 1) + \frac{f''(1)}{2!}(x + 1)^2 + \frac{f'''(1)}{3!}(x + 1)^3 + \frac{f''''(1)}{4!}(x + 1)^4 \ & = 3  2(x + 1) + \frac{10}{2!}(x + 1)^2  \frac{24}{3!}(x + 1)^3 + \frac{24}{4!}(x + 1)^4 \ & = 3  2(x + 1) + 5(x + 1)^2  4(x + 1)^3 + (x + 1)^4 \end{align*} ]
So, the Taylor series for ( f(x) = x^4  x^2 + 1 ) at ( c = 1 ) is:
[ 3  2(x + 1) + 5(x + 1)^2  4(x + 1)^3 + (x + 1)^4 ]
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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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