Question

find the average rate of change of the function over the given interval. ( p(\theta) = \theta^3 - 6\theta^2 + 3\theta; 6,7 ) ( \frac{delta p}{delta \theta} = square ) (simplify your answer.)

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( P(\theta) \) over the interval \([a, b]\) is given by \(\frac{\Delta P}{\Delta \theta}=\frac{P(b)-P(a)}{b - a}\). Here, \( a = 6 \), \( b=7 \), and \( P(\theta)=\theta^{3}-6\theta^{2}+3\theta \).

Step2: Calculate \( P(7) \)

Substitute \( \theta = 7 \) into \( P(\theta) \):
\[

$$\begin{align*} P(7)&=7^{3}-6\times7^{2}+3\times7\\ &=343-6\times49 + 21\\ &=343-294+21\\ &=49 + 21\\ &=70 \end{align*}$$

\]

Step3: Calculate \( P(6) \)

Substitute \( \theta=6 \) into \( P(\theta) \):
\[

$$\begin{align*} P(6)&=6^{3}-6\times6^{2}+3\times6\\ &=216-6\times36+18\\ &=216 - 216+18\\ &=0 + 18\\ &=18 \end{align*}$$

\]

Step4: Calculate the average rate of change

Using the formula \(\frac{P(7)-P(6)}{7 - 6}\):
\[
\frac{70-18}{7 - 6}=\frac{52}{1}=52
\]

Answer:

\( 52 \)