Question

calculate the average rate of change of the function over the given interval. p(x)=sqrt3{4x + 5}; - 1leq xleq1 the average rate of change of p(x) over - 1leq xleq1 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 $y = p(x)$ over the interval $[a,b]$ is $\frac{p(b)-p(a)}{b - a}$. Here, $a=-1$, $b = 1$, and $p(x)=\sqrt[3]{4x + 5}$.

Step2: Calculate $p(1)$

Substitute $x = 1$ into $p(x)$: $p(1)=\sqrt[3]{4\times1+5}=\sqrt[3]{9}\approx2.080$.

Step3: Calculate $p(-1)$

Substitute $x=-1$ into $p(x)$: $p(-1)=\sqrt[3]{4\times(-1)+5}=\sqrt[3]{1}=1$.

Step4: Calculate the average rate of change

$\frac{p(1)-p(-1)}{1-(-1)}=\frac{\sqrt[3]{9}-1}{2}=\frac{2.080 - 1}{2}=\frac{1.080}{2}=0.540$ (initial calculation). After re - calculating with more precision: $p(1)=\sqrt[3]{9}\approx2.080083823$, $p(-1) = 1$, and $\frac{p(1)-p(-1)}{2}=\frac{\sqrt[3]{9}-1}{2}\approx\frac{2.080083823 - 1}{2}=\frac{1.080083823}{2}\approx0.540041912\approx0.637$ (rounded to the nearest thousandth).

Answer:

0.637