Steps in Solving Optimization Problems: Calculus Notes 4.7: Optimization Problems 4.8: Applications to Business and Economics 4.7: Optimization Problems Steps in Solving Optimization Problems: Understand the problem: What is known? Unknown? Want to find? Draw a Diagram: to visualize and see if things make sense. Introduce Notation: label and define variables. Write equations in terms of one variable: re-write everything into one variable. Find relationships between the variables to simplify into one. Use methods of sections 4.1 and 4.3 to find absolute maximum or minimum value of the function. First Derivative Test for Absolute Extreme Values: Suppose that c is a critical number of a continuous function f defined on an interval. If f ‘ (x) > 0 for all x < c and f ‘ (x) < 0 for all x > c, then f (c) is the absolute maximum value of f. If f ‘ (x) < 0 for all x < c and f ‘ (x) > 0 for all x > c, then f (c) is the absolute minimum value of f.

The Average Cost Function: With C(x)=cost function Calculus Notes 4.7: Optimization Problems 4.8: Applications to Business and Economics 4.8: Applications to Business and Economics The Average Cost Function: With C(x)=cost function x is the units of a certain product. If average cost is a minimum, then marginal cost = average cost. Demand Function: With p(x)=price per unit that the company can charge if it sells x units. R(x) is the revenue function (or sales function). Marginal revenue function is R ‘ (x). Profit Function: With P(x) is the profit function. The marginal profit function is P ‘ (x). If the profit is a maximum, then marginal revenue = marginal cost.

Calculus Notes 4.7: Optimization Problems 4.8: Applications to Business and Economics Read Sections 4.7 and 4.8. Use them to help you try the problems for the assignment. Summarize the main concepts for 4.7 and the main concepts for 4.8 for next class. Have the above summary for 4.7 and the summary for 4.8 ready to share in groups by next class. PS 4.7 pg.283 #1, 2, 7, 14, 34, 46 (6) PS 4.8 pg.293 #1, 6, 9, 12, 20 (5)