Find the Taylor series expansion for f(x) =sin 2x at x = pi (write the series with at least three nonzero terms). From part (a), find the 5^th degree polynomial P_5 (x) to approximate f(x). Approximate f(3) using P_5(x) above.
The Correct Answer and Explanation is :
Step 1: Taylor Series Expansion
The Taylor series expansion of ( f(x) ) about ( x = a ) is:
[
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n
]
For ( f(x) = \sin(2x) ), expand at ( x = \pi ).
Derivatives:
- ( f(x) = \sin(2x) ), so ( f(\pi) = \sin(2\pi) = 0 ).
- First derivative: ( f'(x) = 2\cos(2x) ), so ( f'(\pi) = 2\cos(2\pi) = 2 ).
- Second derivative: ( f”(x) = -4\sin(2x) ), so ( f”(\pi) = -4\sin(2\pi) = 0 ).
- Third derivative: ( f”'(x) = -8\cos(2x) ), so ( f”'(\pi) = -8\cos(2\pi) = -8 ).
- Fourth derivative: ( f^{(4)}(x) = 16\sin(2x) ), so ( f^{(4)}(\pi) = 16\sin(2\pi) = 0 ).
- Fifth derivative: ( f^{(5)}(x) = 32\cos(2x) ), so ( f^{(5)}(\pi) = 32\cos(2\pi) = 32 ).
Taylor Series:
Using the formula:
[
f(x) = f(\pi) + f'(\pi)(x-\pi) + \frac{f”(\pi)}{2!}(x-\pi)^2 + \frac{f”'(\pi)}{3!}(x-\pi)^3 + \ldots
]
Substituting derivatives at ( x = \pi ):
[
f(x) \approx 0 + 2(x-\pi) + 0 + \frac{-8}{6}(x-\pi)^3 + 0 + \frac{32}{120}(x-\pi)^5
]
[
f(x) \approx 2(x-\pi) – \frac{4}{3}(x-\pi)^3 + \frac{4}{15}(x-\pi)^5
]
Step 2: Fifth-Degree Polynomial ( P_5(x) )
The fifth-degree polynomial is:
[
P_5(x) = 2(x-\pi) – \frac{4}{3}(x-\pi)^3 + \frac{4}{15}(x-\pi)^5
]
Step 3: Approximation of ( f(3) )
Substitute ( x = 3 ):
[
P_5(3) = 2(3-\pi) – \frac{4}{3}(3-\pi)^3 + \frac{4}{15}(3-\pi)^5
]
Numerically calculating:
[
3 – \pi \approx -0.1416
]
[
P_5(3) \approx 2(-0.1416) – \frac{4}{3}(-0.1416)^3 + \frac{4}{15}(-0.1416)^5
]
[
P_5(3) \approx -0.2832 – 0.000377 + 0.000012
]
[
P_5(3) \approx -0.2836
]
Explanation (300 Words)
The Taylor series expands a function ( f(x) ) about a point ( x = a ), allowing approximation through polynomials. Each term depends on derivatives of ( f(x) ) evaluated at ( x = a ), ensuring local accuracy. For ( f(x) = \sin(2x) ), we derive its Taylor series about ( x = \pi ) using successive derivatives. Odd derivatives contribute non-zero terms, reflecting ( \sin(2x) )’s periodic nature, while even derivatives vanish because ( \sin(2\pi) = 0 ).
The resulting series up to the fifth degree, ( P_5(x) ), includes terms up to ( (x-\pi)^5 ). The series approximates ( f(x) ) closely near ( x = \pi ), with error decreasing as higher-order terms are included.
To approximate ( f(3) ), substitute ( x = 3 ) into ( P_5(x) ). Numerical substitution involves calculating powers of ( (3-\pi) \approx -0.1416 ). The result, ( P_5(3) \approx -0.2836 ), provides a close approximation to ( f(3) ), as higher-degree terms contribute minimally.
This method demonstrates how Taylor series simplify complex trigonometric functions to polynomials, useful in physics, engineering, and numerical computation where exact values are challenging to compute.