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HW01 Average Rate of Change: Problem 6

Let $f(x) = x^2 - 3x$.

Find the average rate of change of $f(x)$ from $x = 2$ to $x = 5$.

This problem asks for the average rate of change of the function $f(x) = x^2 - 3x$ between two points $x = 2$ and $x = 5$.

Solution:

The formula for the average rate of change of a function $f(x)$ from $x = a$ to $x = b$ is:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

Substitute $f(x) = x^2 - 3x$, $a = 2$, and $b = 5$.

1. Calculate $f(2)$: $f(2) = 2^2 - 3(2) = 4 - 6 = -2$

2. Calculate $f(5)$: $f(5) = 5^2 - 3(5) = 25 - 15 = 10$

3. Find the average rate of change: $\text{Average Rate of Change} = \frac{f(5) - f(2)}{5 - 2} = \frac{10 - (-2)}{3} = \frac{10 + 2}{3} = \frac{12}{3} = 4$

Final Answer:

The average rate of change of $f(x)$ from $x = 2$ to $x = 5$ is $\boxed{4}$.

Do you have any questions or would you like further details? Here are five related questions you could explore:

1. What is the significance of the average rate of change in real-world contexts?
2. How does the average rate of change compare to the instantaneous rate of change?
3. Can the average rate of change be negative? What would that indicate?
4. How would you compute the average rate of change if the function was more complex (e.g., $f(x) = x^3 - 5x + 2$)?
5. What would happen to the average rate of change if you increased the interval length?

Tip: Always recheck the function values carefully when calculating rates of change to avoid sign errors!