Let’s go through each part of the problem and solve it step-by-step.

Part A: Calculate the average rate of change of $f$ over $[-2.5, 6]$.

The average rate of change of $f$ over an interval $[a, b]$ is calculated as: $\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

From the graph of $f$, find the approximate values of $f(-2.5)$ and $f(6)$, then substitute into the formula above. Let’s say $f(-2.5) \approx 6$ and $f(6) \approx 2$ (you’ll need to verify these values from the graph for precision).

Substitute: $\text{Average rate of change} = \frac{2 - 6}{6 - (-2.5)} = \frac{-4}{8.5} \approx -0.47$

Now, check if the instantaneous rate of change of $f$, given by $f'$, equals this value at any points on the interval. Look at the graph of $f'$ to see where it reaches $-0.47$. Count the points where this happens to answer the question.

Part B: Estimate the equation of a tangent line at $x = 2$ for the graph of $f$.

To estimate the tangent line at $x = 2$:

1. Find $f(2)$ from the graph of $f$ (let's assume $f(2) \approx 3$).
2. Find $f'(2)$ from the graph of $f'$ (suppose $f'(2) \approx -1$).

The equation of the tangent line is: $y - f(2) = f'(2)(x - 2)$

Substitute the values: $y - 3 = -1(x - 2) \Rightarrow y = -x + 5$

Part C: Find the value for each of the following limits:

1. $\lim_{{x \to -1}} \frac{f(x) - f(-1)}{x + 1}$
2. $\lim_{{x \to 3}} \frac{f(x) - f(3)}{x - 3}$

These limits represent the derivatives of $f$ at $x = -1$ and $x = 3$, respectively. Use the graph of $f'$ to determine $f'(-1)$ and $f'(3)$, as these limits, if they exist, would be equal to those values.

Would you like further explanation or details for any part of this solution? Here are some questions for further exploration:

1. What does the average rate of change signify in this context?
2. How does the graph of $f'$ relate to finding instantaneous rates of change?
3. How would you interpret a tangent line in terms of the function $f$?
4. Why is the limit definition of a derivative important for Part C?
5. How would the answer change if the interval in Part A were different?

Tip: For better accuracy, read values directly from the graph and ensure the tangent line's slope matches $f'$ at the chosen point.