AP Calculus BC · Composite, implicit, and inverse-function differentiation
A continuous function f is defined on the closed interval [1,9]
Problem
A continuous function \(f\) is defined on the closed interval \([1,9]\). Selected values of \(f\) are given in the table. \[ \begin{array}{c|ccccc} x & 1 & 3 & 5 & 7 & 9 \\ \hline f(x) & 4 & 7 & 6 & 11 & 5 \end{array} \] Let \(I=\displaystyle\int_{1}^{9}f(x)\,dx\). (a) Approximate \(I\) using a left Riemann sum with four subintervals of equal length. (b) Approximate \(I\) using a right Riemann sum with four subintervals of equal length. (c) Approximate \(I\) using a trapezoidal sum with four subintervals of equal length. (d) Can the table alone be used to determine whether the left Riemann sum in (a) is an overestimate or an underestimate of \(I\)? Give a reason.
Hint
Equal length means \(\Delta x=(9-1)/4=2\). Left uses the left endpoint of each subinterval; right uses the right endpoint; trapezoidal is the average of the two.
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