To find the average rate of change of the function $f(x) = 2x^2 + 5x - 1$ over the interval $-4 \leq x \leq 0$, we use the formula for the average rate of change:

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

where $a = -4$ and $b = 0$.

Step 1: Calculate $f(a) = f(-4)$

$f(-4) = 2(-4)^2 + 5(-4) - 1 = 2(16) - 20 - 1 = 32 - 20 - 1 = 11$

Step 2: Calculate $f(b) = f(0)$

$f(0) = 2(0)^2 + 5(0) - 1 = -1$

Step 3: Apply the formula

$\text{Average rate of change} = \frac{f(0) - f(-4)}{0 - (-4)} = \frac{-1 - 11}{4} = \frac{-12}{4} = -3$

So, the average rate of change is $-3$.

Would you like further clarification or more details?

Related questions:

1. How is the average rate of change related to the slope of a secant line?
2. What is the geometric interpretation of the average rate of change?
3. How does the result change for different intervals of the same function?
4. Can we find the instantaneous rate of change at a specific point using this method?
5. How do the coefficients of the quadratic function affect the rate of change?

Tip:

The average rate of change is the same as finding the slope of the line that passes through two points on the curve of the function.