Question

What is the average rate of change of the function f(x) = (3x-2)(x+5) / x^2+6x+5 between x = -6 and x = -3?

Answer 1

To find the average rate of change of the function \( f(x) = \frac{(3x-2)(x+5)}{x^2+6x+5} \) between \( x = -6 \) and \( x = -3 \), we will use the formula for the average rate of change:

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

where \( a = -6 \) and \( b = -3 \).

First, we need to evaluate \( f(-6) \) and \( f(-3) \).

**Step 1: Calculate \( f(-6) \)**

\[
f(-6) = \frac{(3(-6) - 2)(-6 + 5)}{(-6)^2 + 6(-6) + 5}
\]
Calculating the numerator:
\[
3(-6) - 2 = -18 - 2 = -20
\]
\[
-6 + 5 = -1
\]
Now the numerator is:
\[
(-20)(-1) = 20
\]

Calculating the denominator:
\[
(-6)^2 + 6(-6) + 5 = 36 - 36 + 5 = 5
\]

So,
\[
f(-6) = \frac{20}{5} = 4
\]

**Step 2: Calculate \( f(-3) \)**

\[
f(-3) = \frac{(3(-3) - 2)(-3 + 5)}{(-3)^2 + 6(-3) + 5}
\]
Calculating the numerator:
\[
3(-3) - 2 = -9 - 2 = -11
\]
\[
-3 + 5 = 2
\]
Now the numerator is:
\[
(-11)(2) = -22
\]

Calculating the denominator:
\[
(-3)^2 + 6(-3) + 5 = 9 - 18 + 5 = -4
\]

So,
\[
f(-3) = \frac{-22}{-4} = \frac{22}{4} = 5.5
\]

**Step 3: Calculate the Average Rate of Change**

Now, we have \( f(-6) = 4 \) and \( f(-3) = 5.5 \).

Using the average rate of change formula:

\[
\text{Average Rate of Change} = \frac{f(-3) - f(-6)}{-3 - (-6)} = \frac{5.5 - 4}{-3 + 6} = \frac{1.5}{3} = 0.5
\]

Thus, the average rate of change of the function \( f(x) \) between \( x = -6 \) and \( x = -3 \) is

\[
\boxed{0.5}.
\]