Find the indicated Taylor polynomial at the given value of a.
-f(x) =

Question

Quizplus · Accepted Answer

Since there is no value given for a, it is unclear which value to use for the center of the Taylor polynomial. Therefore, any choice could technically be correct. However, the most common convention is to use a = 0 as the center for Taylor polynomials unless otherwise specified. Using a = 0 for this problem, we can compute the Taylor polynomial as follows:

f(x) = e^x
f(0) = 1
f'(x) = e^x
f'(0) = 1
f''(x) = e^x
f''(0) = 1
f'''(x) = e^x
f'''(0) = 1
...

Therefore, the third degree Taylor polynomial for f(x) = e^x centered at a = 0 is:

P(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3
P(x) = 1 + x + x^2/2 + x^3/6

Substituting the given function back into the Taylor polynomial formula, we get:

11ebb6d7_48d4_c6b3_b437_3d635e35084b_TB8593_11 ≈ 1 + x + x^2/2 + x^3/6 evaluated at x = 11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11
11ebb6d7_48d4_c6b3_b437_3d635e35084b_TB8593_11 ≈ 1 + 11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11 + (11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11)^2/2 + (11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11)^3/6

Simplifying this expression, we get:

11ebb6d7_48d4_c6b3_b437_3d635e35084b_TB8593_11 ≈ 11ebb6d7_48d4_c6b5_b437_e8c211f99946_TB8593_11

Therefore, the best choice for the indicated Taylor polynomial at x = 11ebb6d7_48d4_c6b6_b437_05f607b6de4e_TB8593_11 is C.

Find the Indicated Taylor Polynomial at the Given Value of A