Find The Marginal Revenue Function. R(x) = x(13 – 0.08x) * REF * 39; (x) = Find The Marginal Profit Function If Cost And Revenue Are Given By c(x) = 219 + 0.8x – 1 And R(x) = 5x – 0.03x . P'(X)=0 Find The Average Cost Function If Cost And Revenue Are Given By c(x) = 145 + 7.1x And R(x) = 6x – 0.02 * 2 The Average Cost Function is C(X)= Find The Marginal Average Cost Function
Find the marginal revenue function R(x) = x(13 – 0.08x) R'(x)=
The Correct Answer and Explanation is :
Let’s break down the problem step by step:
- Marginal Revenue Function: You provided the revenue function as: [
R(x) = x(13 – 0.08x)
] First, expand the revenue function: [
R(x) = x \cdot 13 – x \cdot 0.08x = 13x – 0.08x^2
] The marginal revenue function is the derivative of the revenue function, ( R'(x) ): [
R'(x) = \frac{d}{dx}(13x – 0.08x^2)
] To differentiate: [
R'(x) = 13 – 0.16x
] So, the marginal revenue function is: [
R'(x) = 13 – 0.16x
] - Marginal Profit Function: The profit function is the revenue minus the cost: [
P(x) = R(x) – C(x)
] You provided the cost function as: [
C(x) = 219 + 0.8x – 1
] Simplify the cost function: [
C(x) = 218 + 0.8x
] Now, the profit function is: [
P(x) = (13x – 0.08x^2) – (218 + 0.8x)
] Simplifying: [
P(x) = 13x – 0.08x^2 – 218 – 0.8x
]
[
P(x) = 12.2x – 0.08x^2 – 218
] The marginal profit function is the derivative of the profit function, ( P'(x) ): [
P'(x) = \frac{d}{dx}(12.2x – 0.08x^2 – 218)
] Differentiating: [
P'(x) = 12.2 – 0.16x
] So, the marginal profit function is: [
P'(x) = 12.2 – 0.16x
] - Average Cost Function: The average cost function is given by: [
C_{\text{avg}}(x) = \frac{C(x)}{x}
] Using the cost function ( C(x) = 145 + 7.1x ): [
C_{\text{avg}}(x) = \frac{145 + 7.1x}{x}
]
Simplifying: [
C_{\text{avg}}(x) = \frac{145}{x} + 7.1
] - Marginal Average Cost Function: The marginal average cost function is the derivative of the average cost function: [
C_{\text{avg}}'(x) = \frac{d}{dx} \left( \frac{145}{x} + 7.1 \right)
] Differentiating: [
C_{\text{avg}}'(x) = -\frac{145}{x^2}
] So, the marginal average cost function is: [
C_{\text{avg}}'(x) = -\frac{145}{x^2}
]
Final Results:
- Marginal Revenue Function: ( R'(x) = 13 – 0.16x )
- Marginal Profit Function: ( P'(x) = 12.2 – 0.16x )
- Average Cost Function: ( C_{\text{avg}}(x) = \frac{145}{x} + 7.1 )
- Marginal Average Cost Function: ( C_{\text{avg}}'(x) = -\frac{145}{x^2} )
Let me generate a plot for the Marginal Revenue Function ( R'(x) = 13 – 0.16x ).
Here is the plot for the Marginal Revenue Function ( R'(x) = 13 – 0.16x ). It shows how the marginal revenue changes with respect to the quantity (x). As the value of (x) increases, the marginal revenue decreases due to the negative coefficient of (x) in the equation. This reflects diminishing returns, which is a typical behavior in many economic models.
If you need any further explanations or calculations, feel free to ask!
