(a) Find the polynomial P₁(x) = a₀ + a₁x whose value and slope agree with the value and slope of f(x) = cos x at the point x = 0.

(b) Find the polynomial P₂(x) = a₀ + a₁x + a₂ x² whose value and first two derivatives agree with the value and first two derivatives of f(x) = cos x at the point x = 0. This polynomial is called the second-degree Taylor polynomial of f(x) = cos x at x = 0.

(c) Complete the table comparing the values of f(x) = cos x and P₂(x). What do you observe?

(d) Find the third-degree Taylor polynomial of f(x) = sin x at x = 0.

Transcribed Image Text:

X COS X P₂(x) -1.0 -0.1 -0.001 0 0.001 0.1 1.0