To find the rate of change of the function $f(x) = x^2 + 11x$ over the interval $[-1, -1+h]$, we can use the formula for the average rate of change:

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

where $a = -1$ and $b = -1 + h$.

Step 1: Calculate $f(-1)$

Substitute $x = -1$ into the function $f(x)$:

$f(-1) = (-1)^2 + 11(-1) = 1 - 11 = -10$

Step 2: Calculate $f(-1 + h)$

Substitute $x = -1 + h$ into the function $f(x)$:

$f(-1 + h) = (-1 + h)^2 + 11(-1 + h)$

Expanding the terms:

$(-1 + h)^2 = 1 - 2h + h^2$ $11(-1 + h) = -11 + 11h$

So,

$f(-1 + h) = 1 - 2h + h^2 - 11 + 11h = h^2 + 9h - 10$

Step 3: Calculate the Rate of Change

Now, use the average rate of change formula:

$\text{Rate of Change} = \frac{f(-1 + h) - f(-1)}{(-1 + h) - (-1)}$

Substitute the values we found:

$\text{Rate of Change} = \frac{(h^2 + 9h - 10) - (-10)}{h}$

Simplify:

$\text{Rate of Change} = \frac{h^2 + 9h - 10 + 10}{h} = \frac{h^2 + 9h}{h}$

Simplify further:

$\text{Rate of Change} = h + 9$

Final Answer:

The rate of change of the function $f(x) = x^2 + 11x$ over the interval $[-1, -1+h]$ is $h + 9$.

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Would you like to explore this further or have any specific questions? Here are 5 related questions you might find helpful:

1. How does the rate of change $h + 9$ behave as $h$ approaches zero?
2. What does the rate of change tell us about the function's behavior over the interval?
3. How would the rate of change differ for another quadratic function, like $f(x) = x^2 + 5x$?
4. What is the instantaneous rate of change (derivative) of the function $f(x) = x^2 + 11x$ at $x = -1$?
5. How can you interpret the rate of change geometrically on the graph of the function?

Tip: The average rate of change over an interval is equivalent to the slope of the secant line connecting the endpoints of the function on that interval.