calculate the average rate of change of the function over the given interval. f(x)=\\sqrt3{x}, 3\\leq…

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.)

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.)

Answer

Answer:

0.0547

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

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