In Business, the following functions are important. Revenue function = (price per unit) . (quantity of units) Symbols: Cost function = (average cost per unit) . (quantity of units) Symbols: Profit function = revenue − cost Symbols: Sometimes in a problem some of these functions are given. Note: Do not confuse p and P . The price per unit p is also called the demand function p . Marginal Functions: The derivative of a function is called marginal function. The derivative of the revenue function R(x) is called marginal revenue with notation: The derivative of the cost function C(x) is called marginal cost with notation: The derivative of the profit function P(x) is called marginal profit with notation: Example 1: Given the price in dollar per unit p = −3x2 + 600x , find: (a) the marginal revenue at x = 300 units. Interpret the result. R=p.x C=C.x P=R−C R′(x) = dR dx C′(x) = dC dx P′(x) = dP dx revenue function: R(x) = p . x = (−3x2 + 600x) . x = −3x3 + 600x2 marginal revenue: R′(x) = dR = −9x2 + 1200x dx marginalrevenueat x=300 =⇒ R′(300)= dR =−9(300)2+1200(300)=−450000 dx x=300 Interpretation: If production increases from 300 to 301 units, the revenue decreases by 450 000 dollars. (b) the marginal revenue at x = 100 units. Interpret the result. revenue function: R(x) = p . x = (−3x2 + 600x) . x = −3x3 + 600x2 marginal revenue: R′(x) = dR = −9x2 + 1200x dx marginalrevenueat x=100 =⇒ R′(100)= dR =−9(100)2+1200(100)=30000 dx x=100 Interpretation: If production increases from 100 to 101 units, the revenue increases by 30 000 dollars.
