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CHAPTER 3 SECTION 3.7 OPTIMIZATION PROBLEMS. Applying Our Concepts We know about max and min … Now how can we use those principles?

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Presentation on theme: "CHAPTER 3 SECTION 3.7 OPTIMIZATION PROBLEMS. Applying Our Concepts We know about max and min … Now how can we use those principles?"— Presentation transcript:

1 CHAPTER 3 SECTION 3.7 OPTIMIZATION PROBLEMS

2 Applying Our Concepts We know about max and min … Now how can we use those principles?

3

4 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 principles x 30” 60”

5 Guidelines for Solving Applied Minimum and Maximum Problems

6 Optimization

7 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

8 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.

9 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.

10 Primary Secondary Intervals: Test values: V ’(test pt) V(x) rel max Domain of x will range from x being as small as possible to x as large as possible. Smallest (x is near zero) Largest (y is near zero) Dimensions: 6 in x 6 in x 3 in

11 2.Find the point on that is closest to (0,3).

12 Primary Secondary Intervals: Test values: d ’(test pt) d(x) rel min ***The value of the root will be smallest when what is inside the root is smallest. rel max rel min Minimize distance

13 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?

14 Primary Secondary Intervals: Test values: A ’(test pt) A(x) rel min Smallest (x is near zero) Largest (y is near zero) Print dimensions: 6 in x 4 in Page dimensions: 9 in x 6 in

15 1.Find two positive numbers whose sum is 36 and whose product is a maximum.

16 Primary Secondary Intervals: Test values: P ’(test pt) P(x) rel max

17 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.

18 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?

19 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. area of ends lateral area We need another equation that relates r and h :

20 Example 5: What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area

21 If the end points could be the maximum or minimum, you have to check. 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. 

22 Example #1 A company needs to construct a cylindrical container that will hold 100cm 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

23 Minimizing Cost Domain: r>0

24 Minimizing Cost Concave up – Relative min The container will have a radius of 1.744 cm and a height of 10.464 cm 1.744 0 - - - + + + + + C' changes from neg. to pos.  Rel. min


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