Question

suppose that the total profit in hundreds of dollars from selling x items is given by p(x)=4x² - 7x + 9. complete parts a through d below.
a. find the average rate of change of profit as x changes from 3 to 5.
$ 2500 per item
b. find the average rate of change of profit as x changes from 3 to 4.
$ 2100 per item
c. find and interpret the instantaneous rate of change of profit with respect to the number of items produced when x = 3. (this number is called the marginal profit at x = 3.)
$ 1700 per item
what does this result mean? choose the correct answer below.
a. when 3 items are sold, the profit is decreasing at the rate of $ per item.
b. when items are sold for $, the profit is increasing at the rate of $3 per item.
c. when items are sold for $, the profit is decreasing at the rate of $3 per item.
d. when 3 items are sold, the profit is increasing at the rate of $ per item.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ from $x = a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Calculate average rate of change for part a

Given $P(x)=4x^{2}-7x + 9$. First, find $P(3)$ and $P(5)$.
$P(3)=4\times3^{2}-7\times3 + 9=4\times9-21 + 9=36-21 + 9=24$.
$P(5)=4\times5^{2}-7\times5 + 9=4\times25-35 + 9=100-35 + 9=74$.
The average rate of change from $x = 3$ to $x = 5$ is $\frac{P(5)-P(3)}{5 - 3}=\frac{74 - 24}{2}=\frac{50}{2}=25$. Since $P(x)$ is in hundreds of dollars, the average rate of change is $25\times100 = 2500$ dollars per item.

Step3: Calculate average rate of change for part b

Find $P(4)=4\times4^{2}-7\times4 + 9=4\times16-28 + 9=64-28 + 9=45$.
The average rate of change from $x = 3$ to $x = 4$ is $\frac{P(4)-P(3)}{4 - 3}=\frac{45 - 24}{1}=21$. In hundreds of dollars, it is $21\times100=2100$ dollars per item.

Step4: Find derivative for part c

Differentiate $P(x)=4x^{2}-7x + 9$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $P^\prime(x)=8x-7$.
Evaluate $P^\prime(x)$ at $x = 3$. $P^\prime(3)=8\times3-7=24 - 7=17$. In hundreds of dollars, it is $17\times100 = 1700$ dollars per item.
The instantaneous rate of change (marginal profit) at $x = 3$ means that when 3 items are sold, the profit is increasing at the rate of $1700$ dollars per item.

Answer:

a. $2500$ dollars per item
b. $2100$ dollars per item
c. $1700$ dollars per item; D. When 3 items are sold, the profit is increasing at the rate of $\$1700$ per item.