Solutions

Estimate graphically $A(f)$ and $\xi$, such that $A(f) = f(\xi)$ for the following functions:

• $f(t) = (7 - 2t)$ on $[-2,3]$.
  The graph of $f$ is a straight line segment.
  $A(f)$ should be the value of $f$ half-way along the line segment, so should be $f(0.5) = 6$.
• $f(t) = (t^2 + t)$ on $[-1,2]$.
  Eyeballing the graph, which is a segment of a parabola, we estimate $A(f) = 1.5$.
• $f(t) = \frac{1}{t}$ on $[1,2]$.
  Eyeballing the graph, which is a segment of a rectangular hyperbola, we estimate $A(f) = 0.7$.