Question

calculate the average rate of change of the function over the given interval. f(x)=\sqrt3{x}, 3\leq x\leq14. the average rate of change of f(x) over 3\leq x\leq14 is (type an integer or decimal rounded to the nearest thousandth as needed.)

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 3$, $b=14$ and $f(x)=\sqrt[3]{x}=x^{\frac{1}{3}}$.

Step2: Calculate $f(14)$ and $f(3)$

$f(14)=14^{\frac{1}{3}}\approx2.4101$; $f(3)=3^{\frac{1}{3}}\approx1.4422$.

Step3: Apply the formula

$\frac{f(14)-f(3)}{14 - 3}=\frac{2.4101 - 1.4422}{11}=\frac{0.9679}{11}\approx0.0547$.

Answer:

0.0547