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HH_15_2 - † Lecture notes to accompany Section 15.2 of...

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(11/25/08) Math 10C. Lecture Examples. Section 15.2. Optimization Example 1 The function g ( x , y ) = 2x 2 + y 2 - 2xy - 2y has a global minimum. Find it. Answer: The global minimum is - 2. Example 2 Find, without using calculus, the global maximum of f ( x , y , z ) = 1 ( x - y ) 2 + 1 + e - z 2 and where it occurs. Answer: The global maximum is 2 at the points ( x, x, 0) for all x . Example 3 The function M ( x , y ) = - 5y x 2 + y 2 + 1 , whose graph is shown in Figure 1, has a global maximum and a global minimum. What are they and where do they occur? FIGURE 1 Answer: The global maximum is 5 2 at (0 , - 1) and the global minimum is - 5 2 at (0 , 1). Example 3 Suppose that rectangular boxes with no tops (Figure 2) are to be manufac- tured so that each has a volume of 6 cubic feet. The boxes are to be made from material that costs 6 dollars per square foot for the bottoms, 2 dollars per square foot for the fronts and backs, and 1 dollar per square foot for the sides. What dimensions would minimize the cost of manufacturing each box? FIGURE 2 Answer: The boxes should be manufactured to be 1 foot wide, 2 feet long, and 3 feet high.
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Unformatted text preview: † Lecture notes to accompany Section 15.2 of Calculus by Hughes-Hallett et al. 1 Math 10C. Lecture Examples. (11/25/08) Section 15.2, p. 2 Example 5 The function N ( x , y ) = 1 x + 1 y + xy has a minimum for x > , y > . What is it and where does it occur? Answer: The minimum is 3 at (1 , 1). Example 6 What are the values of x and y such that the sum of the squares of the distances from the point ( x , y ) to the four points ( a 1 , b 1 ) , ( a 2 , b 2 ) , ( a 3 , b 3 ) , and ( a 4 , b 4 ) is a minimum? Answer: x = 1 4 ( a 1 + a 2 + a 3 + a 4 ) • y = 1 4 ( b 1 + b 2 + b 3 + b 4 ) (The averages) Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 15.1: Examples 1–6 ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course....
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HH_15_2 - † Lecture notes to accompany Section 15.2 of...

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