Taylor Polynomials: Approximating f(x) and f'(x)

[Hileon]

Let f be a function that has derivatives of all orders for all real numbers. Assume f(1)=3, f'(1)=-2, f"(1)=2, and f'''(1)=4

a. Write the second-degree Taylor polynomial for f about x=1 and use it to approximate f(0.7).
b. Write the third-degree Taylor polynomial for f about x=1 and use it to approximate f(1.2).
c. Write the second-degree Taylor polynomial for f', the derivative of f, and x=1 and use it to approximate f'(1.2).

a. T{2}(x) = f(1) + f'(1)(x-1) + f"(1)(x-1)^2/2!
plug in .7 = 3.51
b. T{3}(x) = f(1) + f'(1)(x-1) + f"(1)(x-1)^2/2! + f"'(1)(x-1)^3/3!
plug in 1.2 = 2.64
c. What do I do for c?

 

[Hurkyl]

Well, what are the 0-th, 1-st, and 2-nd derivatives of f'?

 

[HallsofIvy]

You know how to differentiate a polynomial don't you?

The third order Taylor's polynomial for f is T{3}(x) = f(1) + f'(1)(x-1) + f"(1)(x-1)^2/2! + f"'(1)(x-1)^3/3!= 3- 2(x-1)+2(x-1)²+ (2/3)(x-1)³.

The second order Taylor's polynomial for f' is the derivative of that:
-2+ 4(x-1)+ 2(x-1)². Now put x= 1.2.