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How do you use a Taylor series to expand: #f(x) = x^2 + 2x + 5# about x = 3?

Answer 1

If #f(x) = x^2+2x+5# then #color(white)("XXXXX")##f'(x) = 2x+2# #color(white)("XXXXX")##f''(x)= 2# #color(white)("XXXXX")##f'''(x)" and beyond" = 0#

The Taylor Series about #x=3# is #color(white)("XXXXX")##f(3)+(f'(3))/(1!)(x-3)+(f''(3))/(2!)(x-3)^2#

#color(white)("XXXXX")##= 20 + 8(x-3) + (x-3)^2#

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Answer 2

Evelyn Agard

To expand ( f(x) = x^2 + 2x + 5 ) about ( x = 3 ) using a Taylor series, follow these steps:

Find the derivatives of ( f(x) ) up to the desired order.
Evaluate each derivative at ( x = 3 ).
Use the Taylor series formula to write the expansion.

The Taylor series expansion of ( f(x) ) about ( x = 3 ) will involve the derivatives of ( f(x) ) evaluated at ( x = 3 ), along with terms involving ( (x - 3) ) raised to the appropriate powers.

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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.