Question

Which of the following is the average rate of change of f(x) with respect to x over the interval
[a, a+h]?

• A. $\frac{f\left(a\right)-f(a+h)}{h}$
• B. $\frac{f\left(a\right)-f(a+h)}{t}$
• C. $\frac{f(a+h)-f\left(a\right)}{h}$
• D. $\frac{f(a+h)-f\left(a\right)}{t}$

Answer 1

The correct option is C

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

Rate of change is the change in one quantity with respect to another quantity. Here, we want to measure the change of f(x) with respect to x. Initial value of f(x) in the interval [a, a+h] is f(a). Final value is f(a+h). So, we can say that the change in f(x) is f(a+h)-f(a). Change in x is (a+h) -a = h.

So to measure the change in f(x) with respect to x, we will take the ratio of the changes. Which is nothing but, $\frac{f(a+h)-f\left(a\right)}{h}$This is the change in f(x) for a unit change in x ( that's why we are dividing by h)

So, the average rate of change of a function f(x) with respect to x over an interval [a, a+h] is defined as

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