For the given logarithmic function, find a) the average rate of change for the interval $[2,6]$ and b) the interval $[3,5]$, as well as $c$ ) the rate of change at $x=4$. Round all calculations to the nearest thousandth.
-$f(x)=\ln (x+1)$
A) a) 0.275 b) 0.255 c) 0.2
B) a) 0.212 b) 0.203 c) 0.2
C) a) 0.2 b) 0.255 c) 0.275
D) a) 0.2 b) 0.203 c) 0.212

Question

Quizplus · Accepted Answer

The average rate of change over an interval $[a,b]$ for a function $f(x)$ is given by $\frac{f(b)-f(a)}{b-a}$. For $f(x)=\ln(x+1)$, a) from $x=2$ to $x=6$, $\frac{\ln(7)-\ln(3)}{6-2} = \frac{\ln(7/3)}{4} \approx 0.212$. b) From $x=3$ to $x=5$, $\frac{\ln(6)-\ln(4)}{5-3} = \frac{\ln(3/2)}{2} \approx 0.203$. c) The rate of change at a point for $f(x)$ is given by $f'(x)$, and $f'(x) = \frac{1}{x+1}$, so at $x=4$, $f'(4) = \frac{1}{5} = 0.2$.

For the Given Logarithmic Function, Find A) the Average Rate [2,6][2,6][2,6]

For the Given Logarithmic Function, Find A) the Average Rate $[2,6]$