find the average rate of change of f(x)=-x² + 4x + 1 from x = 3 to x = 6. simplify your answer as much as…

Explanation:

Step1: Encontrar f(6)

Sustituir x = 6 en f(x): [ \begin{align*} f(6)&=-(6)^2 + 4(6)+1\ &=- 36+24 + 1\ &=-11 \end{align*} ]

Step2: Encontrar f(3)

Sustituir x = 3 en f(x): [ \begin{align*} f(3)&=-(3)^2+4(3)+1\ &=-9 + 12+1\ &=4 \end{align*} ]

Step3: Aplicar la fórmula de tasa media de cambio

La fórmula para la tasa media de cambio de una función f(x) de x = a a x = b es $\frac{f(b)-f(a)}{b - a}$. Aquí, a = 3, b = 6, f(6)=-11 y f(3)=4. [ \begin{align*} \frac{f(6)-f(3)}{6 - 3}&=\frac{-11-4}{6 - 3}\ &=\frac{-15}{3}\ &=- 5 \end{align*} ]

Answer:

-5