find the average rate of change of ( f(x)=x^{3}-4x + 5 ) over the following intervals.
(a) from -5 to -2
(b) from -1 to 4
(c) from 4 to 6
(a) the average rate of change from -5 to -2 is ( square )

Question

Accepted Answer

# Explanation:
## Step1: Recall the formula for average rate of change
The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-5$ and $b = - 2$, and $f(x)=x^{3}-4x + 5$.
## Step2: Calculate $f(-5)$
Substitute $x=-5$ into $f(x)$:
$$
\begin{align*}
f(-5)&=(-5)^{3}-4\times(-5)+5\
&=-125 + 20+5\
&=-100
\end{align*}
$$
## Step3: Calculate $f(-2)$
Substitute $x = - 2$ into $f(x)$:
$$
\begin{align*}
f(-2)&=(-2)^{3}-4\times(-2)+5\
&=-8+8 + 5\
&=5
\end{align*}
$$
## Step4: Calculate the average rate of change
Using the formula $\frac{f(b)-f(a)}{b - a}$, with $a=-5$, $b=-2$, $f(a)=-100$, and $f(b) = 5$:
$$
\begin{align*}
\frac{f(-2)-f(-5)}{-2-(-5)}&=\frac{5-(-100)}{-2 + 5}\
&=\frac{5 + 100}{3}\
&=\frac{105}{3}\
&=35
\end{align*}
$$

# Answer:
$35$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the formula for average rate of change

Step2: Calculate \(f(-5)\)

Step3: Calculate \(f(-2)\)

Step4: Calculate the average rate of change

Answer: