Question

Let $l$be the line connecting two points $(a,f(a\left)\right)$and $(b,f(b\left)\right)$on the graph of a function $f$. What does this line $l$have to do with the average rate of change of $f$on the interval $[a,b],$and why?

Short answer

The line can be the tangent line of the function at$x=c$.

Step-by-step solution

Step 1. Given information

A function $f\left(x\right)$.

A line $l$.

Two points$(a,f(a\left)\right),(b,f(b\left)\right)$.

Step 2. Explanation

The average rate of change of the $f$at point $x=a$in the interval $[a,b]$will be :-

$\frac{f\left(b\right)-f\left(a\right)}{b-a}$

If the intervals is very short, that means $x=a$and $x=b$are very close to each other , then $\frac{f\left(b\right)-f\left(a\right)}{b-a}$will represent the slope of the tangent line at $x=c$and hence the line can be the tangent line of the function at $x=c$.