The average rate of change for a function f(x) at two different values of x can be calculated by the following formula:
[tex] \frac{f( x_{2})-f( x_{1})}{ x_{2} - x_{1} } [/tex]
We are to find the find the average rate of change of given function over the range [1,6], so,
[tex] x_{1}=1 \\
x_{2}=6 [/tex]
Using the values in above formula, we get:
[tex] \frac{f(6)-f(1)}{6-1} \\ \\
= \frac{-ln(6)-(-ln(1))}{5} = \frac{-ln(6)-0}{5} \\ \\
= \frac{-ln(6)}{5} [/tex]
Thus the average rate of change for f(x) = -ln(x) for the interval [1,6] is given by the above expression
