Question

What is the average rate of change of the function f(x)=3x²-5 over the interval [2,6]?

12

3

48

24​

Answer 1

24

Explanation:

The rate of change over a function f(x) is given by its first derivative

`f'(x) = (d)/(dx) (f(x))`

`\mathrm{Differentiating\; f(x) = 3x^2- 5 \;w.r.t \;x : }`

`(d)/(dx)\left(3x^2-5\right) = (d)/(dx)\left(3x^2\right)-(d)/(dx)\left(5\right)\\\\= 6x - 0\\\\= 6x\\\\`

`f'(x) = 6x\\\\`

Over the interval [2, 6] the average rate of change would be
`f'(6) - f'(2)`

`f'(6) = 6 \cdot 6 = 36\\\\f'(2) = 6\cdot 2 = 12\\\\f'(6) - f'(2) = 36 - 12 = 24\\\\`

Average rate of change of the function f(x)=3x²-5 over the interval [2,6]

= 24