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Find the Taylor polynomial of degree 4 for x near the point x= π/4 for the function cos(4x).

P₄ (x)=

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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!

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