Q. 4 - Calculus

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?

Step-by-Step Solution

Answer

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

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 $x=c$.