How do you find the third degree Taylor Polynomial of #f(x)=ln(x^2)# at x=1?
Then:
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To find the thirddegree Taylor polynomial of (f(x) = \ln(x^2)) at (x = 1), we need to find the values of the function and its derivatives at (x = 1). Then, we can use the Taylor series formula to construct the polynomial.

Calculate (f(x)) and its derivatives up to the third order. [f(x) = \ln(x^2)] [f'(x) = \frac{d}{dx} \ln(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2}{x}] [f''(x) = \frac{d}{dx} \left(\frac{2}{x}\right) = \frac{2}{x^2}] [f'''(x) = \frac{d}{dx} \left(\frac{2}{x^2}\right) = \frac{4}{x^3}]

Evaluate these derivatives at (x = 1): [f(1) = \ln(1^2) = \ln(1) = 0] [f'(1) = \frac{2}{1} = 2] [f''(1) = \frac{2}{1^2} = 2] [f'''(1) = \frac{4}{1^3} = 4]

Use the Taylor series formula to construct the polynomial: [P_3(x) = f(1) + f'(1)(x1) + \frac{f''(1)}{2!}(x1)^2 + \frac{f'''(1)}{3!}(x1)^3]

Substitute the values: [P_3(x) = 0 + 2(x1)  \frac{2}{2!}(x1)^2 + \frac{4}{3!}(x1)^3]
Simplify: [P_3(x) = 2(x1)  (x1)^2 + \frac{4}{3}(x1)^3]
This is the thirddegree Taylor polynomial of (f(x) = \ln(x^2)) at (x = 1).
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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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