The question asks to explain the difference between the average rate of change of $y$ as $x$ changes from $a$ to $b$, and the instantaneous rate of change of $y$ at $x = a$.

Key Differences:

1. Average Rate of Change:

   • This is the slope of the secant line between two points $(a, f(a))$ and $(b, f(b))$ on the graph of $y = f(x)$.
   • The formula for the average rate of change from $x = a$ to $x = b$ is: $\frac{f(b) - f(a)}{b - a}$
   • It represents the average change in $y$ with respect to $x$ over the interval $[a, b]$.

2. Instantaneous Rate of Change:

   • This is the slope of the tangent line to the graph of $y = f(x)$ at $x = a$, representing how fast $y$ is changing at a specific point $x = a$.
   • The formula for the instantaneous rate of change at $x = a$ is the derivative: $\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$
   • This is equivalent to the derivative $f'(a)$, which gives the rate of change at that specific point.

Answer Analysis:

• Option A: Incorrect. The formula given for the instantaneous rate of change is wrong. It uses $\frac{f(b) + f(a)}{b + a}$, which is not correct for either the instantaneous or the average rate of change.

• Option B: Correct. It correctly states that the average rate of change is $\frac{f(b) - f(a)}{b - a}$, and the instantaneous rate of change is found by taking the limit as $b$ approaches $a$, which is the derivative.

• Option C: Incorrect. It repeats the wrong formula $\frac{f(b) + f(a)}{b + a}$ for the average rate of change.

• Option D: Incorrect. It correctly states the instantaneous rate of change, but the explanation for the average rate of change being the limit as $b \to a$ is wrong, as that is the definition of the instantaneous rate of change.

Thus, Option B is the correct answer.

Would you like further details or clarification?

Follow-up Questions:

1. How do we interpret the instantaneous rate of change in real-world scenarios?
2. What is the difference between a secant line and a tangent line in terms of rate of change?
3. How does the concept of a derivative relate to the idea of the instantaneous rate of change?
4. Can the average rate of change ever equal the instantaneous rate of change? Under what conditions?
5. How do you compute the average rate of change if the function $f(x)$ is not given explicitly?

Tip:

When calculating the average rate of change, think of it as finding the slope between two points, while the instantaneous rate of change is the slope of a single point, calculated using limits.