a) Explanation Given: The marginal revenue is given by R ′ ( x ) = − 3 x 2 + 4 x + 32 Where x is the level of production. R ( 0 ) = 0 Concept used: Integration of a function can be defined by anti derivative of that function. For any function f ( x ) = b x + C , derivative of the function can be evaluated as f ′ ( x ) = b d d x ( x ) + d d x ( C ) = b ( 1 ) + 0 = b Therefore, antiderivative of b is given by ∫ b d x = b x + C Here, b is any constant and C is integrating constant. Formula used: If Marginal revenue of a commodity is R ′ ( x ) , then the demand function is calculated as p ( x ) = R ( x ) x = ∫ R ′ ( x ) d x x Formula to evaluate the integral of the form ∫ a x n d x is shown below. ∫ a x n d x = a ∫ x n d x = a x n + 1 n + 1 + C [ n ≠ − 1 ] Linear property of inderfinite integral are as follows: ∫ [ a f ( x ) ± b g ( x ) ] d x = a ∫ f ( x ) d x ± b ∫ g ( x ) d x Calculation: Now, consider the given function for marginal revenue. R ′ ( x ) = − 3 x 2 + 4 x + 32 Therefore, the total revenue function for producing x units is as shown below b) To determine To calculate: The level of production for maximum revenue and the market price per unit at the level of production.

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 5.1, Problem 42PS

To determine

To calculate:The demand function p(x) .

To determine

To calculate:The level of production for maximum revenue and the market price per unit at the level of production.

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Chapter 5.1, Problem 41PS

Chapter 5.1, Problem 43PS

Chapter 5 Solutions

Calculus

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The demand function p ( x ) .

Solution Summary: The author explains the formula to calculate the demand function p(x).

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To calculate:The demand function p(x) .

To calculate:The level of production for maximum revenue and the market price per unit at the level of production.

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Chapter 5 Solutions