MTH 251 – Differential Calculus Chapter 4 – Applications of Derivatives Section 4.6 Applied Optimization Copyright © 2010 by Ron Wallace, all rights reserved.

Optimization Problems Application problems that lead to finding the maximum or minimum value of a function over an interval.  Minimize cost  Maximize profit  Minimize materials  Maximize volume  Etc………… Note: Although the function may be defined over a larger domain, the interval for the problem may be restricted due to the conditions of the application.

Optimization Problems: Procedure 1.Read and understand the problem. 2.Identify known and unknown values.  Recommended: Draw and label a diagram.  There will be (at least) 2 unknowns … An independent variable The value (dependent variable) to be optimized 3.Write an equation relating the two variables.  Dependent variable = f(Independent variable)  Specify the domain. 4.Determine the critical points and endpoints. 5.Evaluate the function at the points determined in step 4 to find the optimal value.

Optimization Problems: Example 1 A box with a top is to be made out of a 20” by 30” rectangular piece of cardboard by cutting out six squares (see diagram). How large should the cutout squares be to obtain a box with the largest possible volume? BottomTop 20” 30” x V = x [20 − 2x] [(30 − 3x)/2] = 3x 3 − 60x x x  (0, 10) V’ = 9x 2 − 120x = 0 = 3 (x − 10) (3x − 10) = 0 x = 3 1/3 in V = 444 4/9 in 3 100

Optimization Problems: Example 2 A cylindrical container must hold 16 ounces of liquid (~29 in 3 ). The top of the container costs twice as much as the sides and the bottom. What should be the dimensions of the container in order to minimize the cost of the container? h r 50 2

More examples ………. Students should read through the examples in the book (pages 263 – 268). Consider even problems on pages 268 – 273.