First of all, I'm new to calculus, so please excuse me if my question sounds silly.
I have a graph of the function $f(x)= \ln(x)$, and I want to compute the average rate of change of f on the interval $[1, 6]$, and then draw a straight line representing its slope.
The problem is, when I try to find the average rate of change, I get a decimal slope, and don't know know to add this slope line to the graph, or where to put it.
These were my steps to finding the average rate of change:
$$\frac{f(6) – f(1)} 5 =
\frac{\ln(6) – \ln(1)} 5 =
\frac{\ln(5)}5 = 0.32188$$
How do I finish this problem, and where do I draw this slope line?
Best Answer
First of all $\ln 6 - \ln 1 = \ln 6 - 0 = \ln 6$. So the average rate of change is given by $$\dfrac{\ln 6 - \ln 1}{6 - 1} = \dfrac {\ln 6}{5}\approx 0.358.$$
Now, use the points $(1, \ln 1) = (1, 0)$ and $(6,\ln 6)$ to draw your line (the line that passes through those two points). The average rate of change on the interval is the slope of this line.