Let's analyze the graph and the quantities labeled as I, II, III, and IV to match them with the given expressions in the dropdown list.

The expressions in the dropdowns are:

1. $f(4)$
2. $f(4) - f(2)$
3. $\frac{f(5) - f(2)}{5 - 2}$ (This is the average rate of change from $x = 2$ to $x = 5$)
4. $f'(3)$ (This is the derivative at $x = 3$, which gives the slope of the tangent line at that point)

Analyzing the graph:

1. Label I: This appears to be the vertical distance from the curve to the tangent line at $x = 4$, which looks like it corresponds to $f(4)$.

2. Label II: This vertical distance seems to be the difference between $f(4)$ and $f(2)$, which corresponds to $f(4) - f(2)$.

3. Label III: This large vertical distance from the curve to the x-axis at $x = 4$ looks like the value $f(4)$, but more specifically, it's referencing $f(4)$, not just a difference, so it could match $f'(3)$.

4. Label IV: This looks like the horizontal difference from $x = 4$ to $x = 5$, which represents the rate of change or slope between these points. So, this is likely $\frac{f(5) - f(2)}{5 - 2}$.

Matching:

• $f(4)$ (a) matches Label I
• $f(4) - f(2)$ (b) matches Label II
• $\frac{f(5) - f(2)}{5 - 2}$ (c) matches Label IV
• $f'(3)$ (d) matches Label III

Would you like further details or clarifications on these explanations?

Here are 5 related questions that could help you practice more:

1. What is the geometric meaning of the derivative $f'(x)$?
2. How do you calculate the average rate of change between two points on a function?
3. How would you graphically interpret the difference $f(x_1) - f(x_2)$?
4. Why is the slope of the tangent line important in calculus?
5. How can you use secant lines to estimate a derivative?

Tip: The derivative at a point gives you the slope of the tangent line to the curve at that point, which is a key concept in calculus.