Final answer:
To find the Taylor polynomial of degree 4 for cos(4x) near x=π/4, we use the Taylor polynomial formula and substitute the values of the function and its first four derivatives at x=π/4. The polynomial is P₄(x) = cos(π/4) + (-4sin(π/4))(x - π/4) + (-16cos(π/4))(x - π/4)²/2! + (64sin(π/4))(x - π/4)³/3! + (256cos(π/4))(x - π/4)⁴/4!
Step-by-step explanation:
To find the Taylor polynomial of degree 4 for cos(4x) near the point x = π/4, we need to first find the values of the function and its derivatives at that point. The formula for the Taylor polynomial is:
P₄(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + f''''(a)(x - a)⁴/4!
In this case, since we are finding the polynomial of degree 4, we'll need the values of the function and its first four derivatives at x = π/4.
cos(4x) has the following derivatives:
f(x) = cos(4x)
f'(x) = -4sin(4x)
f''(x) = -16cos(4x)
f'''(x) = 64sin(4x)
f''''(x) = 256cos(4x)
Substituting these values into the Taylor polynomial formula, we get:
P₄(x) = cos(π/4) + (-4sin(π/4))(x - π/4) + (-16cos(π/4))(x - π/4)²/2! + (64sin(π/4))(x - π/4)³/3! + (256cos(π/4))(x - π/4)⁴/4!