4.5 Optimization Problems

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Presentation transcript:

4.5 Optimization Problems Steps in solving Optimization Problems Understand the Problem Ask yourself: What is unknown? What are the given quantities? What are the given conditions? Draw a Diagram Introduce Notation Assign a symbol to the quantity that is to be maximized or minimized (let’s call it Q for now) Also select symbols (a,b,c,…,x,y) for other unknown quantities and label the diagram with the symbols It may help to use initials as suggestive symbols – for example, A for area, h for height, t for time Express Q in terms of the other symbols from Step 3

Steps in solving Optimization Problems (cont.) 5. If Q has been expressed as a function of more than one variable in Step 4 Use the given information to find relationships (in the form of equations) among these variables Then use these equations to eliminate all but one of the variables in the expression for Q Thus Q will be expressed as a function of one variable x, say Q = f (x) . Write the domain of this function Use the methods of Sections 4.1 and 4.3 to find the absolute minimum or maximum value of f . In particular, if the domain of f is a closed interval, then the Closed Interval Method can be used.

A Classic Optimization 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 = wl

A Classic Optimization 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?

Another Optimization Problem 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

Another Optimization Problem What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area