Question

1. revenue function: $r(x) = 40x - x^2$ find marginal revenue at $x = 8$. interpret.

Explanation:

Step1: Recall marginal revenue formula

Marginal revenue is the derivative of the revenue function. So, we first find the derivative of \( R(x) = 40x - x^2 \). The power rule for differentiation is \( \frac{d}{dx}(x^n)=nx^{n - 1} \).
For \( R(x)=40x - x^2 \), the derivative \( R^\prime(x)=\frac{d}{dx}(40x)-\frac{d}{dx}(x^2) \).
Using the power rule, \( \frac{d}{dx}(40x)=40\times1\times x^{1 - 1}=40 \) and \( \frac{d}{dx}(x^2)=2x^{2 - 1}=2x \). So, \( R^\prime(x)=40 - 2x \).

Step2: Evaluate derivative at \( x = 8 \)

Substitute \( x = 8 \) into \( R^\prime(x) \). So, \( R^\prime(8)=40-2\times8 \).
First, calculate \( 2\times8 = 16 \), then \( 40 - 16 = 24 \).

Interpretation:

The marginal revenue at \( x = 8 \) is 24. This means that when the number of units sold is 8, the revenue will increase by approximately 24 dollars for each additional unit sold (or the revenue from selling the 9th unit is approximately 24 dollars, assuming the change in quantity is 1 unit).

Answer:

The marginal revenue at \( x = 8 \) is \(\boldsymbol{24}\). The interpretation is that when 8 units are sold, the revenue increases by approximately 24 dollars for each additional unit sold.