Derivatives of Logarithmic Functions • Find dy/dx 1. 2. 3. 4.
Derivatives of Exponential Functions Find dy/dx 1. 2. 3.
Applied Maxima and Minima • Maximizing Revenue • Minimizing Average Cost • Profit Maximization • Elasticity of Demand
Maximizing Revenue 1. The demand equation for a manufacturer’s product is p = 20 - 0.25q, 0 q 80 where q is the number of units and p is the price per unit. At what value of q will there be maximum revenue? What is the maximum revenue? 2.The demand equation for a monopolist’s product is p = -5q + 30. At what price will revenue be maximized?
Minimizing Average Cost 1.A manufacturer’s total cost function is given by c = q2/4 + 3q + 400 where c is the total cost of producing q units. At what level of output will average cost per unit be a minimum? 2.A manufacturer finds that the total cost c of producing a product is given by the cost function c = 0.05q2 + 5q + 500 At what level of output will average cost per unit be a minimum?
TUGAS 1. For Monopolist’s product, the demand function is p = 72 - 0.04q and the cost function is c = 500 + 30q. At what level of output will profit be maximized? At what price does this occur, and what is the profit? 2. For a monopolist’s product, the demand equation is p = 42 - 4q, and the average cost function is = 2 + 80/q. Find the profit maximizing price
Profit Maximization 3. Suppose that the demand equation for a monopolist’s product is p = 400 - 2q and the average cost function is = 0.2q+4+400/q, where q is number of units and both p and are expressed in dollars per unit a.Determine the level of output at which profit is maximized b.Determine the price at which maximum profit occurs c.Determine the maximum profit
Elasticity of Demand • Economists measure how a change in the price of a product will affect the quantity demanded • The ratio of the resulting percentage change in quantity demanded to a given percentage change in price: % change in quantity % change in price
Continue …... • If p = f(q) is a differentiable demand function, the point elasticity of demand denoted by:
Continue …... 1. When || > 1, demand is elastic 2. When || = 1, demand has unit elasticity 3. When || < 1, demand is inelastic
Continue …... • Example 1 a. Find the point elasticity of demand for the demand function p = 1200 - q2 at q = 10. b. Determine the point elasticity of demand equation q = p2 - 40p + 400 (where q > 0) at p = 15.
Continue …... • If demand is elastic, then < -1, so 1 + 1/ > 0 total revenue increases if demand elastic • If demand is inelastic, then > -1, so 1 + 1/ < 0 total revenue decreases if demand inelastic