CHAPTER 3 SECTION 3 7 OPTIMIZATION PROBLEMS Applying
CHAPTER 3 SECTION 3. 7 OPTIMIZATION PROBLEMS
Applying Our Concepts • We know about max and min … • Now how can we use those principles?
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 60” principles x 30”
Guidelines for Solving Applied Minimum and Maximum Problems
Optimization
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
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.
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.
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 Smallest (y is near zero) (x is near zero) Intervals: Test values: V ’(test pt) V(x) rel max Dimensions: 6 in x 3 in
2. Find the point on that is closest to (0, 3).
2. Find the point on Primary that is closest to (0, 3). Minimize distance Secondary ***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
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?
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 Smallest Largest (x is near zero) (y is near zero) Intervals: Test values: Print dimensions: 6 in x 4 in Page dimensions: 9 in x 6 in A ’(test pt) A(x) rel min
1. Find two positive numbers whose sum is 36 and whose product is a maximum.
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
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?
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. We need another equation that relates r and h: area of ends lateral area
Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area
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
Example #1 • A company needs to construct a cylindrical container that will hold 100 cm 3. 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. r h
Minimizing Cost Domain: r>0
Minimizing Cost Concave up – Relative min ------ +++++ 0 1. 744 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
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