CHAPTER 3 SECTION 3.7 OPTIMIZATION PROBLEMS

1. CHAPTER 3SECTION 3.7OPTIMIZATION PROBLEMS

2. Applying Our Concepts • We know aboutmax and min … • Now how can we use thoseprinciples?

4. 60” x 30” Use the Strategy • What is the quantity to be optimized? • The volume • What are the measurements (in terms of x)? • What is the variable which will manipulated to determine the optimum volume? • Now use calculus principles

5. Guidelines for Solving Applied Minimum and Maximum Problems

6. Optimization

7. Optimization Maximizing or minimizing a quantity based on a given situation Requires two equations: Primary Equation what is being maximized or minimized Secondary Equation gives a relationship between variables

8. 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. To find the maximum (or minimum) value of a function:

9. 1. An open box having a square base and a surface area of 108 square inches is to have a maximum volume. Find its dimensions.

10. 1. An open box having a square base and a surface area of 108 square inches is to have a maximum volume. Find its dimensions. Primary Secondary Domain of x will range from x being as small as possible to x as large as possible. Largest (y is near zero) Smallest (x is near zero) Intervals: Test values: V ’(test pt) V(x) rel max Dimensions: 6 in x 6 in x 3 in

11. 2. Find the point on that is closest to (0,3).

12. 2. Find the point on that is closest to (0,3). Minimize distance Secondary Primary ***The value of the root will be smallest when what is inside the root is smallest. Intervals: Test values: d ’(test pt) d(x) rel min rel max rel min

13. 2. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the print be to use the least paper?

14. 2. A rectangular page is to contain 24 square inches of print. The margins at the top and bottom are 1.5 inches. The margins on each side are 1 inch. What should the dimensions of the print be to use the least paper? Primary Secondary Largest (y is near zero) Smallest (x is near zero) Intervals: Test values: Print dimensions: 6 in x 4 in A ’(test pt) A(x) Page dimensions: 9 in x 6 in rel min

15. 1. Find two positive numbers whose sum is 36 and whose product is a maximum.

16. 1. Find two positive numbers whose sum is 36 and whose product is a maximum. Primary Secondary Intervals: Test values: P ’(test pt) P(x) rel max

17. 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?

18. 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?

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

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

21. 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. If the end points could be the maximum or minimum, you have to check. p

22. r h Example #1 • A company needs to construct a cylindrical container that will hold 100cm3. The cost for the top and bottom of the can is 3 times the cost for the sides. What dimensions are necessary to minimize the cost.

23. Minimizing Cost Domain: r>0

24. 1.744 0 Minimizing Cost Concave up – Relative min - - - - - - + + + + + C' changes from neg. to pos.  Rel. min The container will have a radius of 1.744 cm and a height of 10.464 cm