# Math Insight

### Applet: Predicting position from incomplete data

The map at the left shows Mike's position at $t=0$ by a open blue circle at the intersection of Main and First. The prediction of Mike's position at time $t=3$ is shown by the red X. As time $t$ changes, the gray filled circle shows the predicted position at time $t$. The graph at the right shows the predicted position as a function of $t$ (green line or curve), with the animated gray circle showing position at time $t$. The vertical axis is numbered according to the street numbers. The starting point is shown by a horizontal blue line at 1 (for first street). The predicted final position at time $t=3$ is shown by the horizontal red line. Mike's predicted position is altered depending on whether or not it is based on the information he was running north or slowing down (check the corresponding boxes). Mike's actual position is revealed by checking the “show actual” box. The moving cyan diamonds show his position at time $t$; black X at left and horizontal black line at right show his final position at time $t=3$.

Mike's actual position is given by \begin{align*} f(t) = 1 + 2t - \frac{t^2}{4} - \frac{t^3}{10}+ \frac{t^4}{40} \end{align*} for $0 < t < 3$. The three predictions are based on subsets of the data $f(0)=1$, $f'(0)=2$, and \$f''(0)=-1/2.

Applet file: predicting_position_incomplete_data.ggb