Question

In the above figure, slope of the secant will give the rate of change of f(x) at x = $\frac{(a+b)}{2}$

• A. True
• B. False

Answer 1

The correct option is B

False

Let’s find the slope of the secant to decide if the given statement is correct. The two end points of the given secant are (a, f(a) ) and (b, f(b)).

The slope of this line is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ is

If you remember how we defined average rate of change, you can easily find that the slope is nothing but the average rate of change of f(x) with respect to x over the interval [a,b].

The rate of change at $x=\frac{(a+b)}{2}$

${lim}_{h\to 0}=\frac{f(\frac{(a+b)}{2}+h)-f\left(\frac{(a+b)}{2}\right)}{h}$

This is not equal to average rate of change for any ‘a’ and ‘b’, even though it could give the average rate of change for some f(x), a and b

So, the given statement is false