How do you find the third degree Taylor Polynomial of #f(x)=ln(x^2)# at x=1?

Answer 1

#ln(x^2)approx2(x-1)-(x-1)^2+2/3(x-1)^3#

The Taylor polynomial for #f# centered at #x=c# is given by
#T(x)=sum_(n=0)^oo(f^((n))(c)(x-c)^n)/(n!)#
Since we want a third degree polynomial, we will extend to the #(x-c)^3# term (we need to find three derivatives):
#f(x)=ln(x^2)=2ln(x)" "=>" "f(1)=2ln(1)=0#
#f'(x)=f^((1))(x)=2/x" "=>" "f^((1))(1)=2#
#f^((2))(x)=-2/x^2" "=>" "f^((2))(1)=-2#
#f^((3))(x)=4/x^3" "=>" "f^((3))(1)=4#

Then:

#ln(x^2)approxf(c)+(f^((1))(1)(x-c))/(1!)+(f^((2))(1)(x-c)^2)/(2!)+(f^((3))(1)(x-c)^3)/(3!)#
#ln(x^2)approx0+(2(x-1))/1+(-2(x-1)^2)/2+(4(x-1)^3)/(3!)#
#ln(x^2)approx2(x-1)-(x-1)^2+2/3(x-1)^3#
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Answer 2

To find the third-degree 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.

  1. 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}]

  2. 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]

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

  4. Substitute the values: [P_3(x) = 0 + 2(x-1) - \frac{2}{2!}(x-1)^2 + \frac{4}{3!}(x-1)^3]

Simplify: [P_3(x) = 2(x-1) - (x-1)^2 + \frac{4}{3}(x-1)^3]

This is the third-degree Taylor polynomial of (f(x) = \ln(x^2)) at (x = 1).

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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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