Let's walk through the questions step by step from the image provided:

(a) Find the average rate of change of $f$ on the interval $[10, 60]$.

The average rate of change of a function $f$ on the interval $[a, b]$ is calculated by: $\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$ From the graph:

• $f(60)$ appears to be 800.
• $f(10)$ is approximately 500.

Now we calculate: $\frac{f(60) - f(10)}{60 - 10} = \frac{800 - 500}{60 - 10} = \frac{300}{50} = 6$ The correct answer is $6$, and this was marked as correct.

(b) Identify an interval on which the average rate of change of $f$ is 0.

The average rate of change is 0 if the function does not change over the interval (i.e., it's constant). From the graph, the function is flat between $x = 10$ and $x = 50$, meaning the slope is 0 in that region.

The correct interval is $[10, 50]$, which was selected and marked correct.

(c) Compute the following:

$\frac{f(40) - f(0)}{40 - 0}$ From the graph:

• $f(40)$ appears to be approximately 500.
• $f(0)$ is approximately 500.

Thus: $\frac{f(40) - f(0)}{40 - 0} = \frac{500 - 500}{40 - 0} = \frac{0}{40} = 0$ This is marked as incorrect in the problem, but the calculation is actually correct. The value is $0$.

What does this value represent geometrically?

Since the rate of change is 0, it means that the slope of the line segment from $(0, f(0))$ to $(40, f(40))$ is 0. The correct interpretation geometrically is that it represents "the slope of the line segment from $(0, f(0))$ to $(40, f(40))$." This was marked as correct.

Do you have any questions or need further details on any of these solutions?

5 related questions:

1. How is the average rate of change different from the instantaneous rate of change?
2. What does a zero average rate of change imply about the function over the interval?
3. How can we visually identify where the average rate of change is zero on a graph?
4. What role do the endpoints of an interval play in determining the average rate of change?
5. How would you compute the instantaneous rate of change from a graph?

Tip:

When calculating the average rate of change, always ensure you have accurately read the values of $f(a)$ and $f(b)$ from the graph or function before performing the computation.