Warm up 9/10/14 a) Find all relative extrema using the 2 nd derivative test: b) Find any points of inflection and discuss the concavity of the graph.

3.7 Modeling and Optimization Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999

3.7 Optimization Problems One of the most common applications of calculus involves the determination of minimum and maximum values. Consider how frequently you hear or read terms such as…

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose? There must be a local maximum here, since the endpoints are minimums.

A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

To find the maximum (or minimum) value of a function: 1 Write it in terms of one variable. 2 Find the first derivative and set it equal to zero. 3 Check the end points if necessary.

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? We can minimize the material by minimizing the area. area of ends lateral area We need another equation that relates r and h :

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area

If the end points could be the maximum or minimum, you have to check those also. Notes: If the function that you want to optimize has more than one variable, use substitution to rewrite the function. If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check by using either the 1st or 2 nd derivative test.

You Try: Find two positive numbers such that the sum of the first number squared and the second is 27 and the product is a maximum.

3.7 Optimization Problems A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with maximum volume? Solution

3.7 Optimization Problems Diagram

3.7 Optimization Problems

Work

3.7 Optimization Problems Diagram

3.7 Optimization Problems Table

3.7 Optimization Problems INT. TEST-212 F’NEG.POS.NEG.POS. CONCL.DECR.INCR.DECR.INCR. Critical #’s

3.7 Optimization Problems

Note: 0 yields a relative maximum, there is no absolute maximum since the domain is the entire real line.

Homework: Day 1: Pg odd & 29

Calculus HWQ 9/11/14 What are the dimensions of the square- based, open-top box that has a volume of 20 cubic inches and the smallest surface area?

Day 2 Problems

Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least wire?

3.7 Optimization Problems Write y and z in terms of x.

3.7 Optimization Problems

It could be the proportion from &$##!

3.7 Optimization Problems Obviously, 320 is a common factor Factorable

3.7 Optimization Problems

Solution

3.7 Optimization Problems Diagram You could use all or none of the wire for the square.

You Try: A rancher has 200 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum?

3.7 Optimization Problems HW Day 2 MMM pgs