Chapter 5.1, Problem 25E

Investigate the reason for the name extrapolation. Assume that $F\left(h\right)$ is an $n\text{th}$ order formula for approximating a quantity $Q$, and consider the points $\left(K{h}^2,F\left(h\right)\right)$ and $\left(K{\left(h/2\right)}^2,F\left(h/2\right)\right)$ in the $xy$

-plane, where error is plotted on the $x$

-axis and the formula output on the $y$

-axis. Find the line through the two points (the best functional approximation for the relationship between error and $F$

). The $y$

-intercept of this line is the value of the formula when you extrapolate the error to zero. Show that this extrapolated value is given by formula (5.15).