Thank you for the correction. Let's work with the function $f(x) = 3000 + 15x + 0.2x^2$ and redo the calculations based on this function.

1. Average Rate of Change

To find the average rate of change of $f(x)$ between $x = 10$ and $x = 15$:

$\text{Average Rate of Change} = \frac{f(15) - f(10)}{15 - 10}$

First, calculate $f(10)$ and $f(15)$:

$f(10) = 3000 + 15(10) + 0.2(10)^2 = 3000 + 150 + 20 = 3170$ $f(15) = 3000 + 15(15) + 0.2(15)^2 = 3000 + 225 + 45 = 3270$

Then, compute the average rate of change:

$\text{Average Rate of Change} = \frac{3270 - 3170}{5} = \frac{100}{5} = 20$

2. Instantaneous Rate of Change

The instantaneous rate of change is found by taking the derivative of $f(x)$:

$f'(x) = \frac{d}{dx}[3000 + 15x + 0.2x^2] = 15 + 0.4x$

Now, evaluate the derivative at $x = 10$ and $x = 15$:

At $x = 10$:

$f'(10) = 15 + 0.4(10) = 15 + 4 = 19$

At $x = 15$:

$f'(15) = 15 + 0.4(15) = 15 + 6 = 21$

3. Rate of Change Equal to 0 or 1000

To find when the rate of change equals 0 or 1000:

1. When $f'(x) = 0$: $15 + 0.4x = 0$ $0.4x = -15 \Rightarrow x = \frac{-15}{0.4} = -37.5$

2. When $f'(x) = 1000$: $15 + 0.4x = 1000$ $0.4x = 985 \Rightarrow x = \frac{985}{0.4} = 2462.5$

Summary:

• Average rate of change between $x = 10$ and $x = 15$ is 20.
• Instantaneous rate of change at $x = 10$ is 19 and at $x = 15$ is 21.
• Rate of change equals 0 at $x = -37.5$ and equals 1000 at $x = 2462.5$.

Would you like any more details or have any questions?

Related Questions:

1. How does changing the coefficients of a quadratic function affect its rate of change?
2. What are practical examples of interpreting the instantaneous rate of change in real life?
3. How do you graphically interpret the average rate of change on the curve?
4. What does the point where the rate of change equals zero represent on the graph?
5. How would these calculations change if $f(x)$ had a cubic term?

Tip:

Always verify the units and context when interpreting the rate of change, especially in applied problems, as it gives meaning to the numbers calculated.