Problem 867

Find the average rate of change of the function over the given intervals.

f (x) = 3x³ + 3,

a) [6,8],

b) [-4,4]

Solution:-

a) The average rate of change of a function f(x) over the interval

[x₁ , x2] is as follows.

$\frac{\Delta y}{\Delta x}=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}$

For the function f(x) = 3x³ + 3 and interval [6,8], the average rate of change is as follows.

$\frac{\Delta y}{\Delta x}=\frac{[(3)(8)^{3}+3]-[(3)(6)^3 +3]}{8-6}$

$\frac{\Delta y}{\Delta x}=$ 444

Thus , the average rate of change of the function 3x³ + 3 over the interval [2,4]is 444.

b) For the function 3x³ + 3 and interval [-4,4]

$\frac{\Delta y}{\Delta x}=\frac{[(3)(4)^{3}+3]-[(3)(-4)^3 +3]}{4-(-4)}$

$\frac{\Delta y}{\Delta x}=$ 48

Thus, the average rate of change of the function 3x³ + 3 over the interval [-2 , 2] is 48.