A function f has derivatives of all orders for all real numbers, x. Assume f(2) = −3, f ′(2) = 5, f ″(2) = 3 and the third derivative of f at x = 2 is −8.

Question

Write the 3rd degree Taylor polynomial for f about x = 2 and use it to approximate f(1.5).

Answer 1

P₃(x) = - 3 + 5(x - 2) + 3(x - 2)²/2! - 8(x - 2)³/3! = - 3 + 5(x - 2) + 3(x - 2)²/2 - 4(x - 2)³/3;

f(1.5) ≈ P₃(1.5) = - 3 - 2.5 + 0.375 + 0.1667 = -4.9583