4_7 - x using the relationships between them Then rewrite...

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Section 4.7: Optimization Problems Instructor: Ms. Hoa Nguyen ([email protected]) In this section, we solve such problems as: Maximizing areas, volumes, and profits Minimizing distances, times, and costs In solving such practical problems, the greatest challenge is often to convert the word problem into a mathematical optimization problem - by setting up the function that is to be maximized or minimized. The First Derivative Test Suppose c is a critical number of a continuous function f 1. f 0 changes from positive (for ALL x < c ) to negative (for ALL x > c ) at c f has a absolute maximum at c . 2. f 0 changes from negative (for ALL x < c ) to positive (for ALL x > c ) at c f has a absolute minimum at c . 3. f 0 does not change sign at c f has NO maximum or minimum at c . Steps in Solving Optimization Problems 1. Read the problem to find the given quantities, the unknown variables, and the given conditions. Drawing a diagram (if possible) will help. 2. Form an equation of the unknowns which is to be maximized or minimized. 3. Transform all the unknowns into one unknown
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Unformatted text preview: x , using the relationships between them. Then rewrite the equation in Step 2 as the equation of x , for example, Q = f ( x ). • 4. Write the domain of the function f . • 5. Find the first derivative f of f . • 6. Find critical numbers of f by setting f ( x ) = 0. • 7. If the domain of f is a closed interval [ a,b ], use the Closed Interval Method to find the absolute maximum or minimum value of f (i.e., compare the values of f at the critical numbers in Step 6 and at the endpoints a,b ). • 8. Otherwise, use the First Derivative Test to find the absolute minimum or maximum value of f . Note An alternative method for solving optimization problems is to use implicit differentiation. Examples Example 1 of Section 4.7 (textbook): Example 2 of Section 4.7 (textbook): Example 3 of Section 4.7 (textbook): Example 4 of Section 4.7 (textbook): Example 5 of Section 4.7 (textbook): 1...
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