Section 12.1 Techniques for Finding Derivative

1. Section 12.1 Techniques for Finding Derivative

2. Constant Rule Power Rule Sum and Difference Rule

3. Examples Find y’ for the function.

4. Constant Times a Function Examples: Find dy/dx for the function.

5. Marginal Functions Example: The total cost in dollars incurred each week for manufacturing qrefrigerators is given by the total cost function: C(q) =8000 + 200q– 0.25q2 What is the actual cost incurred for manufacturing the 251st refrigerator?

6. Marginal Cost • Marginal cost is the cost incurred in producing an additional unit of a certain item given that the plant is already at a certain level of operation. • Mathematically, marginal cost is the rate of change of the total cost function with respect to x • derivative of the cost function. • If C(x) is a total cost function, then the derivative C’(x) is called the marginal cost function.

7. Example The total cost in dollars incurred each week for manufacturing qrefrigerators is given by the total cost function: C(q) =8000 + 200q– 0.25q2 Find the marginal cost for producing 251 refrigerators.

8. Marginal Revenue / Profit If R(x) is a revenue function, then the derivative R’(x) is called the marginal revenue function. If P(x) is a profit function, then the derivative P’(x) is called the marginal profit function. Recall R(q) = pqor R(x) = px and P = R - C

9. Example The total cost C(q) =8000 + 200q– 0.25q2 Suppose the demand equation for the refrigerators each week is given by q = 9000 – 5p. Find the marginal revenue for the production level of 200 units. 400 units.