Question

find the average rate of change of the function $f(x) = \frac{2}{2x + 3}$, on the interval $x \in -2, -1$.
average rate of change =
give an exact answer.

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( f(x) \) on the interval \([a, b]\) is given by \(\frac{f(b)-f(a)}{b - a}\). Here, \( a=-2 \), \( b = - 1\) and \( f(x)=\frac{2}{2x + 3}\).

Step2: Calculate \( f(-2) \)

Substitute \( x=-2 \) into \( f(x) \):
\( f(-2)=\frac{2}{2\times(-2)+3}=\frac{2}{-4 + 3}=\frac{2}{-1}=-2 \)

Step3: Calculate \( f(-1) \)

Substitute \( x = - 1\) into \( f(x) \):
\( f(-1)=\frac{2}{2\times(-1)+3}=\frac{2}{-2 + 3}=\frac{2}{1}=2 \)

Step4: Calculate the average rate of change

Using the formula \(\frac{f(b)-f(a)}{b - a}\), substitute \( a=-2 \), \( b=-1 \), \( f(-2)=-2 \) and \( f(-1)=2 \):
\(\frac{f(-1)-f(-2)}{-1-(-2)}=\frac{2-(-2)}{-1 + 2}=\frac{2 + 2}{1}=\frac{4}{1}=4\)

Answer:

\( 4 \)