Looking at the graph of $f(x)$, we are asked to estimate the values of the derivatives at various points based on the slopes of the tangent lines to the graph. Let's work through each point:

Given:

$f'(0) = 2$, meaning the slope of the tangent line at $x = 0$ is already provided.

$f'(1)$:

At $x = 1$, the graph shows a clear slope that we can estimate by observing how steep the curve is. The slope appears to be negative but not too steep. An estimate for this would likely be around $-1$.

$f'(1.5)$:

At $x = 1.5$, the slope appears to be approximately zero since it is at a local minimum where the curve flattens out. Therefore, $f'(1.5) \approx 0$.

$f'(2)$:

At $x = 2$, there is a sharp corner in the graph, indicating that the function is not differentiable at this point. This confirms the statement that $f(x)$ is not differentiable here.

$f'(3)$:

At $x = 3$, the graph is sloping upwards, and the slope seems relatively steep. An estimate would be around $2$, similar to the slope at $x = 0$.

$f'(4)$:

At $x = 4$, the slope of the graph appears to be zero again, as this is another local maximum. Therefore, $f'(4) \approx 0$.

$f'(5)$:

At $x = 5$, the slope is negative and moderately steep. An estimate for this would be around $-1$.

Summary of estimates:

• $f'(0) = 2$ (given)
• $f'(1) \approx -1$
• $f'(1.5) \approx 0$
• $f'(2)$: Not differentiable
• $f'(3) \approx 2$
• $f'(4) \approx 0$
• $f'(5) \approx -1$

Would you like more details or have any questions on this?

Follow-up questions:

1. What is the significance of a function being not differentiable at a point?
2. How can we determine whether a point is a local maximum or minimum from the derivative?
3. What does the second derivative tell us about the concavity of the function?
4. How would the derivative values change if the graph were shifted upwards or downwards?
5. How can we visually determine a sharp corner or cusp on a graph?

Tip:

When estimating the derivative from a graph, always think about the slope of the tangent line: positive slopes indicate positive derivatives, zero slope indicates zero derivative, and negative slopes indicate negative derivatives.