Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Law of large numbers wikipedia , lookup

Mathematics of radio engineering wikipedia , lookup

Infinitesimal wikipedia , lookup

Location arithmetic wikipedia , lookup

Real number wikipedia , lookup

Large numbers wikipedia , lookup

Elementary arithmetic wikipedia , lookup

Transcript

1 Lesson Plan #27 Date: Wednesday November 14th, 2011 Class: AP Calculus Topic: Optimization Problems Aim: How do we solve optimization problems? Objectives: 1) Students will be able to solve problems where they have to maximize or minimize a value. HW# 27: 1) The sum of one number and two times a second number is 24. What numbers should be selected so that their product is as large as possible? 2) The product of two positive numbers is 192. What numbers should be chosen so that the sum of the first plus three times the second is a minimum? Do Now: You run a small tutoring school. The graph at right represents the amount of profit you take in per week depending on the number of students you have. Based on the graph, approximately how many students should you take in to maximize your profit? With respect to the topics we have discussed, what do we call the location where this maximum occurs? In general, what do we do to find this relative maximum? Procedure: Write the Aim and Do Now Get students working! Take attendance Give back work Go over the HW Collect HW Go over the Do Now Example: Find two positive numbers that minimize the sum of twice the first number plus the second if the product of the two numbers is 288. Solution: 𝑥 = 1𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑦 = 2𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑆 = 2𝑥 + 𝑦, 𝑥𝑦 = 288 Solve for 𝑥 𝑥 = 288 𝑦 Substitute in 𝑆 = 2𝑥 + 𝑦 𝑆 = 2 ( ′ Differentiate 𝑆 = − 576 𝑦2 288 𝑦 )+𝑦 𝑆 = 576 𝑦 +𝑦 +1 Set equal to zero to find critical numbers − 576 𝑦2 + 1 = 0 𝑦 = 24, 𝑥 = 12 2 Example 1: The difference between two numbers is 50. Find the two numbers so that their product is as small as possible Example 2: Find two positive numbers whose sum is 110 and whose product is a maximum Example 3: The sum of one positive number and twice a second positive number is 100. Find the two numbers so that their product is a maximum Example 4: A rectangle has a perimeter of 100 feet. What length and width should it have so that its area is maximum? Example 5: Find the dimensions of a cone (height and radius) of maximum volume if the diameter and height of the cone total 24 cm. Example 6: 1 The graph of 𝑦 = − 𝑥 + 2 encloses a region with the x-axis and y-axis in the first quadrant. A rectangle in the enclosed region 2 1 has a vertex at the origin and the opposite vertex on the graph of f 𝑦 = − 𝑥 + 2. Find the dimensions of the rectangle so that its 2 area is maximum. 3 Sample Test Questions: 1) A line is drawn through the point (1,2) forming a right triangle with the positive x- and y-axes. The slope of the line forming the triangle of least area is 1 A) -1 B) -2 C) -4 D) E) -3 2 2) The area of the largest isosceles triangle that can be drawn with one vertex at the origin and with the others on a line parallel to and above the x-axis and on the curve y 27 x 2 is A) 12 3 B) 27 C) 24 3 D) 54 E) 108 Free Response: 3) If the perimeter of an isosceles triangle is 18cm , find the maximum area of the triangle. 4 If enough time: 4) A net enclosure for golf practice is open at one end, as shown in the figure. Find the dimensions that require the 1 3 least amount of netting if the volume of the enclosure its to be 83 cubic meters. x x y 5) An indoor physical fitness room consists of a rectangular region with a semicircle at each end. If the perimeter of the room is to be a 200-meter running track, find the dimensions, to the nearest tenth of a meter, of the rectangle that will make the area of the rectangular region as large as possible. 6) Find the dimensions of the rectangle with the greatest area that can be inscribed in the semicircle y 9 x2