A Scientist Collects the Following Data and Plots It in the Coordinate

Essay

A scientist collects the following data and plots it in the coordinate plane. $\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 & 5 \\\hline y & 4 & 10 & 8 & 6 & 14 & 10 & 12 \\\hline\end{array}$
(a) Use Simpson's rule with n = 6 to estimate the area under the graph of a continuous function drawn through these points.(b) If it is known that $- 4 \leq f ^ { ( 4 ) } ( x ) \leq 1$ for all x, estimate the error involved in the approximation in part (a).(c) How large do we have to choose n so that the approximation $S _ { n }$ (Simpson's Rule) to the integral is accurate to within 0.001?