Question

Find the fifth degree Taylor polynomial approximation of T _{5}(x) centered at a=0 to the function f(x)=cos(x);f(x)=sin(x);f(x)=e^{x}.

Answer 1

To find the fifth degree Taylor polynomial approximation of the functions f(x) = cos(x), f(x) = sin(x), and f(x) = e^x centered at a=0, we use the Taylor series expansion formula.

Step-by-step explanation:

To find the fifth degree Taylor polynomial approximation of the functions f(x) = cos(x), f(x) = sin(x), and f(x) = e^x centered at a=0, we can use the Taylor series expansion formula. The formula for the nth degree Taylor polynomial is given by:

T_n(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + f^n(a)(x-a)^n/n!

For f(x) = cos(x), the fifth degree Taylor polynomial centered at a=0 would be:

T_5(x) = 1 - x^2/2 + x^4/24

For f(x) = sin(x), the fifth degree Taylor polynomial centered at a=0 would be:

T_5(x) = x - x^3/6 + x^5/120

For f(x) = e^x, the fifth degree Taylor polynomial centered at a=0 would be:

T_5(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120

Answer 2

The question ask to find the fifth grade of Taylor Polynomial approximation of the said equation and also its function. Based on my own calculation and following some theories, the best answer would be P5(x) = 1+x +x^2/2+x^3/3!+x^4/4!+x^5/5!. I hope you are satisfied with my answer and feel free to ask for more