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This Lesson (OVERVIEW of lessons on finding the minimum/maximum of a quadratic function) was created by by ikleyn(52750)
  About ikleyn: OVERVIEW of lessons on finding the minimum/maximum of a quadratic functionBelow is the list of my lessons in this site on finding the minimum/maximum of a quadratic function - HOW TO complete the square to find the minimum/maximum of a quadratic function - Briefly on finding the minimum/maximum of a quadratic function - HOW TO complete the square to find the vertex of a parabola - Briefly on finding the vertex of a parabola - A rectangle with a given perimeter which has the maximal area is a square - A farmer planning to fence a rectangular garden to enclose the maximal area - A farmer planning to fence a rectangular area along the river to enclose the maximal area - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area - Finding the maximum area of the window of a special form - Using quadratic functions to solve problems on maximizing revenue/profit - When is the best time to sell a pig ? - Find the point on a given straight line closest to a given point in the plane - Minimal distance between sailing ships in a sea - Advanced lesson on finding minima of (x+1)(x+2)(x+3)(x+4) The list of lessons with short annotationsHOW TO complete the square to find the minimum/maximum of a quadratic functionIn this lesson the rule is proved on finding the minimum/maximum of a quadratic function y =
Problem 1. Find the minimum of the quadratic function f(x) = Problem 2. Find the maximum of the quadratic function f(x) = Problem 3. Prove that polynomial p(x) = Problem 4. Prove that polynomial p(x) = Problem 5. Prove that polynomial p(x) = Briefly on finding the minimum/maximum of a quadratic function The same rule repeated one more time. HOW TO complete the square to find the vertex of a parabola In this lesson the rule is proved on finding the vertex of a parabola y =
Problem 1. Find the vertex of the parabola f(x) = Problem 2. Find the maximum of the quadratic function f(x) = Problem 3. Prove that polynomial p(x) = Briefly on finding the vertex of a parabola The same rule repeated one more time. A rectangle with a given perimeter which has the maximal area is a square Theorem. Among rectangles with a given perimeter, a square has the maximal area. Example 1. A rectangle has a perimeter of 100 meters. What are the dimensions of the sides if the area is a maximum? Problem 1. A piece of wire 20 inches long is to be cut into a rectangle. How long should each side of the rectangle be to maximize the area of the rectangle? Problem 2. A farmer has 280 feet of fence to enclose a rectangular garden. What dimensions for the garden give the maximum area? A farmer planning to fence a rectangular garden to enclose the maximal area Problem 1. A farmer has 280 feet of fence to enclose a rectangular garden. What dimensions for the garden give the maximum area? A farmer planning to fence a rectangular area along the river to enclose the maximal area Problem 1. A farmer plans to fence a rectangular grazing area along a river with 300 yards of fence. What is the largest area he can enclose? Problem 2. A rancher needs to enclose two adjacent rectangular​ corrals, one for cattle and one for sheep. If the river forms one side of the corrals and 480 yd of fencing is​ available, find the largest total area that can be enclosed. Problem 3. A farmer wants to fence a field using the river as one side of the enclosed area. The farmer has 2000 feet of fencing and wants to have 5 pens. What are the dimensions of the area if the farmer wants the largest pens possible ? A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area Problem 1. A rancher has 1200 feet of fencing to enclose two adjacent rectangular corrals of equal lengths and widths as shown in the figure. What is the maximum area that can be enclosed in the fencing? Problem 2. A rancher has 216 feet of fencing to enclose two adjacent rectangular corrals. What dimensions will produce the largest total area? What is the maximum total area? Finding the maximum area of the window of a special form Problem 1. A window is to be constructed in the shape of an equilateral triangle on top of a rectangle. If its perimeter is to be 600 cm, what is the maximum possible area of the window? Problem 2. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle). If the perimeter of the window is 32 ft, find the width of the window so that the greatest possible amount of light is admitted. Using quadratic functions to solve problems on maximizing revenue/profit Problem 1. A movie theater holds 1000 people. With the ticket price at $8 during the week, the attendance at the theater has been 200 people. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 50. What ticket price will maximize the revenue? Problem 2. A cable television firm presently serves 6,300 households and charges $14 per month. A marketing survey indicates that each decrease of $1 in the monthly charge will result in 630 new customers. Find the value of maximum monthly revenue. Problem 3. Total profit P is the difference between total revenue R and total cost C. Given the following total-revenue and total-cost functions R(x) = find the total profit, the maximum value of the total profit, and the value of x at which it occurs. Problem 4. Each orange tree grown in California produces 630 oranges per year if not more than 20 trees are planted per acre. For each additional tree planted per acre, the yield per tree decreases by 15 oranges. How many trees per acre should be planted to obtain the greatest number of oranges? Problem 5. A travel agency offers an organization an all-inclusive tour for $800 per person if not more than 100 people take the tour. However, the cost per person will be reduced to $5 for each person in excess of 100. How many people should take the tour. In order for the agency to receive the largest gross revenue, and what is the largest gross revenue? Problem 6. A real estate company owns 228 efficiency apartments, which are fully occupied when the rent is $990 per month. The company estimates that for each $25 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income? When is the best time to sell a pig ? Problem 1. Abby has a pig that presently weighs 200 pounds. She could sell it now for a price of $1.40 per pound. The pig is gaining 5 pounds per week while the price per pound is dropping 2 cents per week. When should Abby sell the pig to get the maximum amount of money for it? What is the maximum profit? Find the point on a given straight line closest to a given point in the plane Problem 1. Find the point on the line y = 4x that is closest to the point P = (1,2). Problem 2. Find the point on the line y = 5x+1 that is closest to the point (3,5). Minimal distance between sailing ships in a sea Problem 1. Ship A sailing due north at 6 km/h sights ship B sailing northwest 4 km away and sailing due east at 5 km/h. How close will the two ships pass if neither alters course or speed? Advanced lesson on finding minima of (x+1)(x+2)(x+3)(x+4) Problem 1. Without the derivative find the value of minima of (x+1)(x+2)(x+3)(x+4). Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I. This lesson has been accessed 2554 times. |
