Hi
Trying to work with confusing Taylor series....any assistance would be much appreciated!!
How can I use T(x)=5-9((x-2)^2)-3((x-2)^3)
to approximate the f(0) ????
I realize that Taylor series f(x)=f(a)+f'(a)(x-a)+f''(a)((x-a)^2)etc...
but the above T(x) is centered at x=2,
BUT I need to know when f(x) is centered at x=0
please assist! thank you!
Hi! I can help you with your question. In order to approximate f(0) using the given Taylor series T(x), you need to shift the center of the series from x = 2 to x = 0.
Let's say g(x) is the Taylor series centered at x = 0. To find g(x), you need to substitute x - a with x - 0, which simplifies to just x. The formula becomes:
g(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)^2 + ...
To find f(0), you need to find the value of g(x) when x = 0. So let's calculate the terms of the Taylor series up to the desired order.
f(0) ≈ g(0) ≈ T(x) ≈ 5 - 9((x - 2)^2) - 3((x - 2)^3)
Substitute x = 0 into T(x):
f(0) ≈ g(0) ≈ T(0) ≈ 5 - 9((0 - 2)^2) - 3((0 - 2)^3)
Evaluate the terms:
f(0) ≈ g(0) ≈ T(0) ≈ 5 - 9(4) - 3(-8)
Simplify:
f(0) ≈ g(0) ≈ T(0) ≈ 5 - 36 + 24
f(0) ≈ g(0) ≈ T(0) ≈ -7
Therefore, the approximation for f(0) using the Taylor series T(x) is -7.
I hope this helps! Let me know if you have any further questions.