To find the average rate of change of the function $y$ over the interval $x = 1$ to $x = 5$, we need to use the formula for the average rate of change:

$\text{Average rate of change} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

However, the values for $y$ at $x = 1$ and $x = 5$ are not provided in the table. Instead, we have values at $x = 0, 2, 4,$ and $6$. To approximate the average rate of change over $x = 1$ to $x = 5$, we can interpolate between the points available in the table.

Step 1: Interpolate to Estimate $y(1)$ and $y(5)$

Estimating $y(1)$:

Since $x = 1$ is between $x = 0$ and $x = 2$, we can use linear interpolation between these two points:

• At $x = 0$, $y = 25$.
• At $x = 2$, $y = 56.25$.

The slope between these points is: $\frac{56.25 - 25}{2 - 0} = \frac{31.25}{2} = 15.625$

Using this slope, we can estimate $y(1)$ as: $y(1) = y(0) + (1 - 0) \times 15.625 = 25 + 15.625 = 40.625$

Estimating $y(5)$:

Since $x = 5$ is between $x = 4$ and $x = 6$, we use linear interpolation here as well:

• At $x = 4$, $y = 126.563$.
• At $x = 6$, $y = 284.766$.

The slope between these points is: $\frac{284.766 - 126.563}{6 - 4} = \frac{158.203}{2} = 79.1015$

Using this slope, we estimate $y(5)$ as: $y(5) = y(4) + (5 - 4) \times 79.1015 = 126.563 + 79.1015 = 205.6645$

Step 2: Calculate the Average Rate of Change from $x = 1$ to $x = 5$

Now, we can find the average rate of change over the interval $x = 1$ to $x = 5$: $\text{Average rate of change} = \frac{y(5) - y(1)}{5 - 1} = \frac{205.6645 - 40.625}{4} = \frac{165.0395}{4} = 41.26$

Thus, the average rate of change on the interval $x = 1$ to $x = 5$ is approximately 41.26 (rounded to the nearest thousandths).

Would you like a more detailed explanation or further assistance?

Here are five related questions:

1. How would we find the exact values if more data points were available for $y$?
2. Can we use a quadratic interpolation instead of linear for potentially better accuracy?
3. How would the calculation differ if we used endpoints outside the interval $x = 1$ to $x = 5$?
4. What is the significance of the average rate of change in practical applications?
5. How does the average rate of change differ from the instantaneous rate of change?

Tip: Interpolation between data points is a useful way to estimate values that lie within the range of known data, especially when actual values are unavailable.