The original question is as follows:
(a) Find the degree 2 interpolating polynomial P2(x) through the points (0,0),(π/2,1), and (π,0).
(b) Calculate P2(π/4), an approximation for sin(π/4).
(c) Use Theorem 3.3 to give an error bound for the approximation in part (b).
(d) Compare the actual error to your error bound.
I solved for part A using Newton's Divided Differences method. This resulted in: $$ \begin{align*} P_2(x) &= f[x_1] + f[x_1 x_2](x-x_1) + f[x_1 x_2 x_3](x-x_1)(x-x_2) \\ &=0 + \frac{2}{\pi} (x-0) + - \frac{4}{\pi^2} (x-0)(x-\frac{\pi}{2}) \\ &= \frac{2}{\pi}x - \frac{4}{\pi^2} x (x-\frac{\pi}{2}) \end{align*} $$
To solve part B, I simply plugged in $\frac{\pi}{4}$. I got $P_2(\frac{\pi}{4}) = \frac{3}{4}$.
Part C is where I am having issues. I'm not sure how to give an error bound, however, the solution key provides this as the solution: $$ \frac{\pi^3}{128} \approx 0.242 $$ Could someone please explain how to get this solution?
I assume to solve part D, then we can do something like this: $$ |\sin(\frac{\pi}{4}) - P_2(\frac{\pi}{4})| = |\frac{\sqrt{2}}{2} - \frac{3}{4}| \approx 0.043 < 0.242 $$