10. (a) Find T4(x), the 4th degree Taylor polynomial for f(x) = sin x centered at a = π/6. (b) Use Taylor’s inequality to estimate the accuracy of the approximation f(x) ≈ T4(x) when 0 ≤ x ≤ π/3. In

Step-by-step explanation:

By the general expresion:

T4(x) = f(a) + (f'(a)/1!)*(x-a) + (f''(a)/2!)*(x-a)² + (f'''(a)/3!)*(x-a)³ + (f''''(a)/4!)*(x-a)^4

Then:

f(x) = sinx ; f'(x) = cosx ; f''(x) = -sinx ; f'''(x) = -cosx ; f''''(x) = sinx

Since:  sin(π/6) = 1/2 ; cos(π/6) =

Thus, in the expresion for T4(x):

Inequality, by the Taylor's theorem (jpg adjunt):

We need to find M in the interval indicate. Of the analysis from the graf of the function (jpg adjunt), we see a good candidate is:

M = f(π/3) = sin(π/3) =

Then, in the Taylor's theorem:

(a)  

(b)  

Step-by-step explanation:

(a) We have that Taylor's polynomial is given by

In this case we have to compute T4, with a = pi\6. Hence we have to calculate the first derivative until the fourth derivative of f(x):

By replacing we have

(b) the Taylor's inequality is given by

where f(x)<=M, Hence we have

HOPE THIS HELPS!

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step-by-step explanation:

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