Question

It is known that f(-8) = -3, f'(-8)= -2, f''(-8) = - 14, f(3)( - 8) = - 4, and f(4)( - 8) = - 13. Find P4(3), the fourth degree Taylor polynomial for f(x) centered at x = - 8. (Start with the lowest degree to the highest degree terms.)

Answer 1

Explanation:

The fourth degree Taylor polynomial for f(x) centered at x = -8 can be expressed as:

P4(x) = f(-8) + f'(-8)(x + 8) + [f''(-8)/2!](x + 8)^2 + [f(3)(-8)/3!](x + 8)^3 + [f(4)(-8)/4!](x + 8)^4

Substituting the given values, we get:

P4(x) = -3 - 2(x + 8) - [14/2!](x + 8)^2 - [(-4)/3!](x + 8)^3 - [(-13)/4!](x + 8)^4

Simplifying, we get:

P4(x) = -3 - 2(x + 8) - 7(x + 8)^2/2 + 2(x + 8)^3/3! - 13(x + 8)^4/4!

Therefore, the fourth degree Taylor polynomial for f(x) centered at x = -8 is:

P4(x) = -3 - 2(x + 8) - 7(x + 8)^2/2 + 2(x + 8)^3/6 - 13(x + 8)^4/24