Optimization

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Optimization

• AP CALCULUS

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Steps

• If appropriate, sketch a diagram of the
  situation.
• Determine restrictions on the domain.
• Use the information from the problem to find an
  equation to be optimized.
• Take the derivative, find where f(x) 0
• Verify that point is relative min or max
• First Derivative Test (pos to neg ?max neg to
  pos ?min
• Second Derivative Test (concave up ?min concave
  down ?max
• Check endpoints of domain (absolute mins or maxs
  could occur here)

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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.

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Minimizing Cost
Domain rgt0
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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
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Problem 14
x
x
Perimeter 50 cm
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Problem 14
x
x
25 x
25 x
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Maximizing
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Example 2

• A bookstore can obtain a certain book from the
  publisher at a cost of 3 per book. The
  bookstore has been selling the book at a price of
  15 per copy. At this price, the store has been
  selling 200 copies per month. Research shows
  that for every 1 reduction in the price of the
  book, sales will increase by 20 books per month.
  At what price should the store sell the book to
  maximize profit?

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Maximizing Profit
Let x each 1 decrease in price.
Profit (Store Price Store Cost)(Units
Sold) Currently 12200 2400
Dx-10 lt x lt 12
Concave Down - Max
Price should be set at 14
Critical Point at x 1
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Find a Function

• Describe the function at the point x3 based on
  the following

12
Find a Function

• Describe the function at the point x5 based on
  the following

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Find a Function

• Given the function is continuous at the point
  x2, sketch a graph based on the following

(2,3)