MTH 251 – Differential Calculus Chapter 4 – Applications of DerivativesOptimization ProblemsOptimization Problems: ProcedureOptimization Problems: ExampleSlide 5More examples ……….MTH 251 – Differential CalculusChapter 4 – Applications of DerivativesSection 4.5Applied Optimization ProblemsOptimization Problems•Application problems that lead to finding the maximum or minimum value of a function over an interval.Minimize costMaximize profitMinimize materialsMaximize volumeEtc…………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 optimized3. 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: ExampleA 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?BottomTop20”30”xV = x [20 − 2x] [(30 − 3x)/2] = 3x3 − 60x2 + 300xx (0, 10)V’ = 9x2 − 120x + 300 = 3 (x − 10) (3x − 10)0lim0Vx0lim10Vx 4.444310 Vx = 3 1/3 inV = 444 4/9 in3Optimization Problems: ExampleA 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 , if the box can be no more than 3” tall??BottomTop20”30”xV = x [20 − 2x] [(30 − 3x)/2] = 3x3 − 60x2 + 300xx (0, 3]V’ = 9x2 − 120x + 300 = 3 (x − 10) (3x − 10)0lim0Vx( )3 441V =x = 3 inV = 441 in3Revised …Note that the critical points 10 and 10/3 are outside the domain.More examples ……….•Students should read through the examples in the book (pages 302 – 308).•Consider even problems on pages 309 –
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