Question

suppose that the total profit in hundreds of dollars from selling x items is given by p(x)=4x^2 - 7x + 9. complete parts a through d below. a. find the average rate of change of profit as x changes from 3 to 5. b. find the average rate of change of profit as x changes from 3 to 4. 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.)

Explanation:

Step1: Recall the formula for the derivative

The derivative of a function $y = ax^{2}+bx + c$ is $y'=2ax + b$. Given $P(x)=4x^{2}-7x + 9$, its derivative $P'(x)$ using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $P'(x)=8x-7$.

Step2: Evaluate the derivative at $x = 3$

Substitute $x = 3$ into $P'(x)$. We have $P'(3)=8\times3-7$.
$P'(3)=24 - 7=17$. Since $P(x)$ is in hundreds of dollars, the instantaneous rate of change (marginal profit) at $x = 3$ is $17\times100=\$1700$ per item.

Answer:

$1700$