Question

For the function f(x) = sqrt(x), how do you find the average rate of change over an interval [a, b]?

A. (f(b) - f(a))/(b - a)
B. (f(b) + f(a))/(b + a)
C. (b - a)/(f(b) - f(a))
D. 1/f'(c), using the Mean Value Theorem, where c is in (a, b).

Answer 1

The average rate of change of the function f(x) = sqrt(x) over the interval [a, b] is calculated using the formula (f(b) - f(a)) / (b - a), which is option A.

Step-by-step explanation:

The function described is f(x) = √x, and we are looking for the average rate of change over an interval [a, b]. The average rate of change of a function can be thought of as the slope of the line that connects the points (a, f(a)) and (b, f(b)) on the graph of the function, which is similar to computing average speed in physics. The correct formula to calculate the average rate of change for any function over an interval is given by option A: (f(b) - f(a)) / (b - a).

Applying this to our function, we get the average rate of change on the interval [a, b] as:

(√b - √a) / (b - a)

Therefore, the correct option for finding the average rate of change of the function f(x) = √x over the interval [a, b] is option A.