Question

given the function $h(x)=-x^{2}+7x + 19$, determine the average rate of change of the function over the interval $1leq xleq10$.

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = h(x)$ over the interval $[a,b]$ is $\frac{h(b)-h(a)}{b - a}$. Here, $a = 1$, $b = 10$, and $h(x)=-x^{2}+7x + 19$.

Step2: Calculate $h(1)$

Substitute $x = 1$ into $h(x)$:
$h(1)=-(1)^{2}+7(1)+19=-1 + 7+19=25$.

Step3: Calculate $h(10)$

Substitute $x = 10$ into $h(x)$:
$h(10)=-(10)^{2}+7(10)+19=-100 + 70+19=-11$.

Step4: Calculate the average rate of change

$\frac{h(10)-h(1)}{10 - 1}=\frac{-11 - 25}{9}=\frac{-36}{9}=-4$.

Answer:

$-4$