4.7 Solving Max-Min Problems

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4.7 Solving Max-Min Problems

1. Read3. Identify the known quantities and the
   unknowns. Use a variable.
2. Identify the quantity to be optimized. Write a
   model for this quantity. Use appropriate
   formulas. This is the primary function.
3. If too many variables are in the primary function
   write a secondary function and use it to
   eliminate extra variables.
4. Find the derivative of the primary function.
5. Set it equal to zero and solve.
6. Reread the problem and make sure you have
   answered the question.

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An open box is to be made by cutting squares from
the corner of a 12 by 12 inch sheet and bending
up the sides. How large should the squares be cut
to make the box hold as much as possible?
Figure 3.43 An open box made by cutting the
corners from a square sheet of tin. (Example 1)
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An open box is to be made by cutting squares from
the corner of a 12 by 12 inch sheet and bending
up the sides. How large should the squares be cut
to make the box hold as much as possible?
Figure 3.43 An open box made by cutting the
corners from a square sheet of tin. (Example 1)
Maximize the volume
V l w h
V (12 2x) (12 2x) x 144x 48x2 4x3
V ? 144 96x 12x2 12(12 8x x2)
12(12 8x x2) 0
(6-x)(2-x) 0
x 6 or x 2
V ? ? -92 24x is negative at x 2. There is
a relative max. Box is 8 by 8 by 2 128 in3.
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Minimizing surface area
Figure 3.46 The graph of A 2? r 2 2000/r is
concave up.
You have been asked to design a 1 liter oil can
in the shape of a right cylinder. What dimensions
will use the least material?
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Figure 3.46 The graph of A 2? r 2 2000/r is
concave up.
You have been asked to design a 1 liter oil (1
liter 1000cm3) can in the shape of a right
cylinder. What dimensions will use the least
material?
Minimize surface area
where
Use the 2nd derivative test to show values give
local minimums.
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4.8 Business Terms
x number of items p unit price C Total cost
for x items R xp revenue for x items
average cost for x units
P R C or xp - C
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The daily cost to manufacture x items is C 5000
25x 2. How many items should manufactured to
minimize the average daily cost.
14 items will minimize the daily average cost.
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4.10 Old problem
Given a function, find its derivative
function
derivative
Inverse problem
Given the derivative, find the function. .
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Find a function that has a derivative y 3x2
The answer is called the antiderivative
You can check your answer by differentiation
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Curves with a derivative of 3x2
Each of these curves is an antiderivative of y
3x2
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Antiderivatives
Derivative
Antiderivative
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Find an antiderivative
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Find antiderivatives
Check by differentiating
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Find an antiderivative
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Trigonometric derivatives
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Derivative
Antiderivative