Question

Find the quadratic polynomial ax^2+bx+c which best fits the function f(x)=8^x at x=0, in the sense that g(0)=f(0), and f'(0)=g'(0), and f''(0)=g''(0).
g(x)=?

Answer 1

F(0) = 8^0 = 1
f '(x) = 8^x * ln8; x= 0 => f '(0) = ln8
f ''(x) = 8^x (ln8)^2 ; x = 0=> f ''(0) = (ln8)^2

g(0) = a(0)^2 + b(0) + c = f(0) = 1 => c = 1

g '(x) = 2ax + b => g '(0) = 2a(0) + b = f '(0) = ln8 => b = ln(8)

g''(x) = 2a => g''(0) = 2a = f''(0) = (ln8)^2 => a = (ln8)^2 / 2

=> g(x) = [(ln8)^2 /2] x^2 + [ln(8)]x + 1