Questions: x y ------ 1 -1 2 1 3 3 4 3.5 5 4.3 6 5 7 6.2 8 7 Compute the average rate of change

Solution Steps

To compute the average rate of change of \( y \) with respect to \( x \) over the given data points, we can use the formula for the average rate of change, which is the change in \( y \) divided by the change in \( x \). Specifically, we will calculate the difference between the \( y \)-values and the \( x \)-values for the first and last data points and then divide these differences.

We are given the following data points: \[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -1 \\ \hline 2 & 1 \\ \hline 3 & 3 \\ \hline 4 & 3.5 \\ \hline 5 & 4.3 \\ \hline 6 & 5 \\ \hline 7 & 6.2 \\ \hline 8 & 7 \\ \hline \end{array} \]

The change in \( y \) (\( \Delta y \)) is calculated as: \[ \Delta y = y_8 - y_1 = 7 - (-1) = 8 \]

The change in \( x \) (\( \Delta x \)) is calculated as: \[ \Delta x = x_8 - x_1 = 8 - 1 = 7 \]

The average rate of change of \( y \) with respect to \( x \) is given by: \[ \text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{8}{7} \approx 1.1429 \]

Final Answer