To calculate the average rate of change of $f(x) = x^2 + 3x$ over a given interval, we use the formula:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}$

Where $a$ and $b$ are the endpoints of the interval.

Part (a) Interval $2 \leq x \leq 3$

Here, $a = 2$ and $b = 3$. We need to calculate $f(2)$ and $f(3)$:

$f(x) = x^2 + 3x$

• $f(2) = 2^2 + 3(2) = 4 + 6 = 10$
• $f(3) = 3^2 + 3(3) = 9 + 9 = 18$

Now, use the formula for the average rate of change:

$\text{Average rate of change} = \frac{f(3) - f(2)}{3 - 2} = \frac{18 - 10}{3 - 2} = \frac{8}{1} = 8$

So, the average rate of change over the interval $2 \leq x \leq 3$ is 8.

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Would you like to proceed with parts (b) and (c)? Or do you need more details on how to compute these values?

Here are 5 related questions:

1. How does the average rate of change relate to the slope of a curve?
2. Can you interpret the average rate of change geometrically?
3. What would happen if the interval was reversed, i.e., $3 \leq x \leq 2$?
4. How can you find the average rate of change over a non-continuous interval?
5. What is the difference between instantaneous and average rate of change?

Tip: For faster calculations, always remember the formula and plug values directly into it—this is especially useful for quadratic functions.