For question 022, we are tasked with determining the graph of the derivative $f'$ given the graph of a function $f$ on the interval $(-2, 2)$.

Analyzing the function $f$:

1. From the graph of $f$, we see that it is piecewise linear. The slope of each segment will give us the derivative $f'$.
2. On $(-2, -1)$, the graph of $f$ is decreasing, meaning $f'$ will be negative here.
3. On $(-1, 0)$, the graph of $f$ is increasing, meaning $f'$ will be positive here.
4. On $(0, 1)$, the graph of $f$ is constant, so the derivative $f'$ will be zero here.
5. On $(1, 2)$, the graph of $f$ is decreasing again, so $f'$ will be negative here.

Identifying the correct derivative:

1. For the interval $(-2, -1)$, the slope is negative, so $f'$ should be below the x-axis in this region.
2. For $(-1, 0)$, the slope is positive, so $f'$ should be above the x-axis in this region.
3. For $(0, 1)$, since the function is constant, $f' = 0$.
4. For $(1, 2)$, the slope is negative again, so $f'$ should be below the x-axis.

Looking at the provided options, option 2 is the correct graph of the derivative. This option shows the correct changes in sign (positive and negative) and zero values corresponding to the flat section of $f$.

For question 023, we are asked to find the graph of the derivative $f'$ of a function on $(-7, 7)$ based on its provided graph.

Analyzing the function $f$:

1. The function starts increasing, meaning $f'$ is positive.
2. Then, the function reaches a maximum at $x = -4$, where $f' = 0$.
3. Between $x = -4$ and $x = 0$, the function decreases, meaning $f'$ is negative.
4. At $x = 0$, the function reaches another minimum, so $f' = 0$.
5. Between $x = 0$ and $x = 2$, the function increases, meaning $f'$ is positive.
6. After $x = 2$, the function decreases again, meaning $f'$ is negative.

The derivative should capture these sign changes and critical points. Based on this reasoning, you can analyze the provided options and select the correct graph of $f'$.

Would you like more details on how to interpret these graphs or any specific part of the process explained further?

Here are some related questions for practice:

1. How does the slope of the graph of $f$ relate to the graph of its derivative $f'$?
2. What happens to the derivative $f'$ at points where $f$ has local maxima or minima?
3. How can you graphically identify intervals where $f$ is increasing or decreasing?
4. What is the relationship between the zeroes of $f'$ and the turning points of $f$?
5. How does the concavity of $f$ relate to the graph of $f'$?

Tip: Always examine how the slope of a function behaves across intervals when determining its derivative.