1 3.7 Optimization Problems Applied Maximum and Minimum Problems minimize cost maximize profit minimize waste least time optimization means finding where some function (model) has its greatest or smallest value. Def.
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3.7 Optimization Problems
Applied Maximum and Minimum Problems
minimize costmaximize profitminimize wasteleast time
optimization means finding where some function (model) has its greatest or smallest value.
Def.
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If f is continuous on the closed interval, then where f has a maximum or minimum must be one of the following:
interior points where f ' = 0,1.
2. interior points where f ' does NOT exist,
3. endpoints of the function's domain.
Concept: (Section 3.1)
none of these points is necessarily the location of an extreme value, but
these are the only candidates!
Note:
1. 2. 3.
critical numbers endpoints
| | | | | |a b a ab b
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Defs.
secondary equation an equation/formula relating the independent variables of the primary equation.
primary equation an equation/formula for the quantity to be optimized.
to reduce the primary equation to one having a single independent variable before attempting to
maximize or minimize it.
Purpose:
feasible domain the values of "x" that make sense in the problem.
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(p 212)
1.
2.
Identify all given quantities and quantities to be determined. When feasible, make a sketch.
3.
Guidelines for Solving Applied Minimum and Maximum Problems
4.
5.
Write a primary equation for the quantity that is to be maximized (or minimized). (A review of several useful formulas from geometry is presented inside the front or back cover.)
Reduce the primary equation to one having a single independent variable. This may involve the use of secondary equations relating the independent variables of the primary equation.
Determine the feasible domain of the primary equation. That is, determine the values for which the stated problem makes sense.
Determine the desired maximum or minimum value by the calculus techniques discussed in Sections 3.1 through 3.4.
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examples:
Find two numbers whose sum is 100 and whose product is a maximum.1.
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Product Design: An opentop box is to be made by cutting small congruent squares from the corners of a 20by25 inch sheet of tin and bending up the sides. Use analytic calculus to determine how large the squares cut from the corners should be to make the box hold as much as possible, the resulting maximum value, and support your answer graphically.
2.
20"
25"
xx
xx x
x
xx
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2. . . . and support your answer graphically.
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[p216 #12]Minimum Distance:3.
f (x) = Find the point on the graph of that is closest to the point (2, 0).x 8
.(2, 0)
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[p218 #30]Area:4.A rectangular page is to contain 36 square inches of print. The margins on each side are to be 11/2 inches. Find the dimensions of the page such that the least amount of paper is used.
11/2"
11/2"
11/2" 11/2"
"You start with a dream. But without hard work, you may end up with nothing but the dream. Hold onto your dreams. They're very important. The ideal situation, however, is to go out and make those dreams come true.
11/2"
11/2"
11/2" 11/2"
"You start with a dream. But without hard work, you may end up with nothing but the dream. Hold onto your dreams. They're very important. The ideal situation, however, is to go out and make those
11/2"
11/2"
11/2" 11/2"
"You start with a dream. But without hard work, you may end up with nothing but the dream. Hold onto your dreams. They're very important. The ideal situation, however, is to go out and make those dreams come true.
11/2"
11/2"
11/2" 11/2"
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Geometry:5. A rectangular is to be inscribed in a semicircle of radius 2. Find the largest area of the rectangle and its dimensions analytically. Support your answer with a graphing utility.
(2, 0)(2, 0)
(0, 2)
.
.
.(2, 0)(2, 0)
(0, 2)
.
.
..(x, y)
(2, 0)(2, 0)
(0, 2)
.
.
..(x, y)
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5. . . . and support your answer graphically.
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Minimum Time:6. Agent 00 is waiting at the edge of a straight canal 1 mile wide, in a motorboat capable of going 40 mph. There is a straight road along the opposite side of the canal where her partner will have a 50mph motorcycle waiting for her wherever she lands. At midnight, she will receive a package to be delivered to a man in a MercedesBenz 5 miles down the road.
Mission: Deliver the package in the shortest possible time. Problem: Where should her partner park the motorcycle?
00
.MB
00A.1 mile
5miles
| |
.C
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6.
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Minimum Cost:7. A powerhouse is on one edge of a straight river and a factory is on the other edge, 100 meters downstream. The river is 50 meters wide. It costs $10 per meter to run electrical cable across the river and $7 per meter on land. How should the cable be installed to minimize the cost?
.F.PH
50 m
100 m| |
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7.
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Assignment:p216217 #1, 3, 5, 7, 9, #1720, 23 #11, 13, 29, 27, #31, 33, 39, 43p217219 #22, 44, 46, 49 #50(a & b), 54
example 4
example 3
example 5
example 1
example 2
examples 6 and 7
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