1. Applications of Extrema Lesson 6.2

2. A Rancher Problem You have 500 feet of fencing for a corral What is the best configuration (dimensions) for a rectangular corral to get the most area One side of the rectangle already has a fence 2

3. Sample Problem Your assistant presents you with a contract for signature Your firm offers to deliver 300 tables to a dealer at $90 per table and to reduce the price per table on the entire order by $0.25 for each additional table over 300 What should you do? Find the dollar total involved in largest (smallest) possible transaction between the manufacturer and the dealer. 3

4. Solution Strategy 1. Read the problem carefully Make sure you understand what is given Make sure you see what the unknowns are From our problem Given 300 tables at $90 per table $0.25 reduction per table on entire order if > 300 Unknowns Largest possible transaction Smallest possible transaction 4

5. Solution Strategy 2. If possible sketch a diagram Label the parts From our problem Not much to diagram … More likely in a problem about the size of a box to minimize/maximize materials or volume 5 x + 3 x 2x

6. Solution Strategy Decide on a variable to be maximized (minimized) Express variable as a function of one other variable Be sure to find function domain From our problem T = transaction amount T = f(x) = ? 6

7. Solution Strategy To analyze the function, place it in Y= screen of calculator Check the table (♦Y) to evaluate the domain and range for setting the graph window 7

8. Solution Strategy 4. Find the critical points for the function View on calculator For our problem Use derivative tests to find actual points 8 Note jump in discontinuous function

9. Solution Strategy 5. If domain is closed interval Evaluate at endpoints, critical points See which value yields absolute max or min For our problem 9 maximum minimum

10. Strategy Review 1. Read carefully, find knowns, unknowns 2. Sketch and label diagram 3. Determine variable to be max/min Express as function of other variable Determine domain 4. Find critical points 5. If domain is closed interval Check endpoints Check critical points 10

11. Practice Problem A fence must be built to enclose a rectangular area of 20,000 ft 2 Fencing material costs $3/ft for the two sides facing north and south It costs $6/ft for the other two sides Find the cost of the least expensive fence 11

12. Assignment Lesson 6.2 Page 383 Exercises 5 – 33 odd 12