2. 4.4 Modeling and Optimization

3. What you’ll learn about Examples from Mathematics Examples from Business and Industry Examples from Economics Modeling Discrete Phenomena with Differentiable Functions Essential Questions How can we use differential calculus to solve optimization problems?

4. Quick Review

5. Strategy for Solving Max-Min Problems

6. Example Inscribing Rectangles 1.A rectangle is to be inscribed under one arch of the sine curve. What is the largest area the rectangle can have, and what dimensions give that area? Graph it to find solutions. The area of the rectangle is A(0.71) = 1.12. The length is 1.72and the height is0.65.

7. Example Inscribing Rectangles 2.Two sides of a triangle have length a and b, and the angle between them is . What value of  will maximize the triangle’s area? A Right Triangle

8. Example Inscribing Rectangles 3.You are a designing a rectangular poster to contain 50 in 2 of printing with a 4-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used?

9. Maximum Profit

10. Example Maximizing Profit 4. The maximum profit occurs at about 3.414.

11. Minimizing Average Cost 5.Using the following equation where x represents thousands of units, determine if there is a production level that minimizes cost. If so, what is it? Marginal cost: Average cost:

12. Pg. 226, 4.4 #1-41 odd