Slide 1 Application of Differentiation

Slide 2 ### Derivatives of Logarithmic Functions

Slide 3 ### Derivatives of Exponential Functions

Find dy/dx

1.

2.

3.

Slide 4 ### Applied Maxima and Minima

- Maximizing Revenue
- Minimizing Average Cost
- Profit Maximization
- Elasticity of Demand

Slide 5 ### 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?

Slide 6 ### 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?

Slide 7 ### 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

Slide 8 ### 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

Slide 9 ### 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

Slide 10 ### Continue …...

- If p = f(q) is a differentiable demand function, the point elasticity of demand denoted by:

Slide 11 ### Continue …...

1. When || > 1, demand is elastic

2. When || = 1, demand has unit elasticity

3. When || < 1, demand is inelastic

Slide 12 Slide 13 ### Elasticity and Revenue

Slide 14