Find the fourth degree Taylor polynomial centered at for the function.

Question

Find the fourth degree Taylor polynomial centered at  for the function. ​  ​

Quizplus · Accepted Answer

To find the fourth degree Taylor polynomial, we need to find the first four derivatives of the function at the given point:

f(x) = 11eac44f_ec36_dc09_bedf_b5fdd0a6ed13_TB7497_11
f'(x) = 0
f''(x) = 0
f'''(x) = 0
f''''(x) = 0

The Taylor polynomial is then given by:

P(x) = f(11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11) + f'(11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)(x-11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11) + (f''(11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)/2!)(x-11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)^2 + (f'''(11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)/3!)(x-11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)^3 + (f''''(11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)/4!)(x-11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11)^4

Since all of the derivatives are zero, the polynomial simplifies to:

P(x) = f(11eac44f_ec36_dc08_bedf_692f51383136_TB7497_11) = 11eac44f_ec36_dc09_bedf_b5fdd0a6ed13_TB7497_11

Therefore, the answer is (C).

Find the Fourth Degree Taylor Polynomial Centered at for the Function