Let’s break down the problem step by step:<\/p>\n\n\n\n
- \n
- Marginal Revenue Function:<\/strong> 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<\/strong> is: [
R'(x) = 13 – 0.16x
]<\/li>\n\n\n\n- Marginal Profit Function:<\/strong> The profit function<\/strong> 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<\/strong> is: [
P'(x) = 12.2 – 0.16x
]<\/li>\n\n\n\n- Average Cost Function:<\/strong> The average cost function<\/strong> 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
]<\/li>\n\n\n\n- Marginal Average Cost Function:<\/strong> The marginal average cost function<\/strong> 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<\/strong> is: [
C_{\\text{avg}}'(x) = -\\frac{145}{x^2}
]<\/li>\n<\/ol>\n\n\n\nFinal Results:<\/h3>\n\n\n\n
- \n
- Marginal Revenue Function:<\/strong> ( R'(x) = 13 – 0.16x )<\/li>\n\n\n\n
- Marginal Profit Function:<\/strong> ( P'(x) = 12.2 – 0.16x )<\/li>\n\n\n\n
- Average Cost Function:<\/strong> ( C_{\\text{avg}}(x) = \\frac{145}{x} + 7.1 )<\/li>\n\n\n\n
- Marginal Average Cost Function:<\/strong> ( C_{\\text{avg}}'(x) = -\\frac{145}{x^2} )<\/li>\n<\/ul>\n\n\n\n
Let me generate a plot for the Marginal Revenue Function<\/strong> ( R'(x) = 13 – 0.16x ).<\/p>\n\n\n\n
Here is the plot for the Marginal Revenue Function<\/strong> ( 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.<\/p>\n\n\n\n
If you need any further explanations or calculations, feel free to ask!<\/p>\n\n\n\n
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-674-1024x815.png\")
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- Marginal Profit Function:<\/strong> ( P'(x) = 12.2 – 0.16x )<\/li>\n\n\n\n
- Marginal Profit Function:<\/strong> The profit function<\/strong> is the revenue minus the cost: [