AD - Optimization Problems Lesson
Optimization Problems
Optimization is the term used to describe the general problem of finding extrema of a function, and the derivative is the tool to accomplish this. Anywhere the derivative of a function is zero is a logical possibility for a maximum or minimum of the function.
Applied Minimum and Maximum Problems
Finding extreme values has enormous practical applications in many areas of life. Just as certain "classic" problems are associated with related rates problems, certain "classic" optimization problem types exist. View each of the four presentations below that address four different classic optimization problems.
Suggestions for Solving Optimization Problems
The greatest challenge in solving practical problems is converting the word problem into a mathematical optimization problem. Steps to follow in solving applied optimization problems are as follows:
1. Read and understand the problem. What is the unknown quantity to be optimized? What are the given quantities? What are the given conditions?
2. Draw a picture. Label any part that may be important to the problem such as the given and required quantities.
3. Introduce variables and notation. Assign a symbol to the quantity to be optimized and list every relation in the picture and in the problem as an equation or algebraic expression.
4. Write an equation for the unknown quantity. If possible, express the unknown as a function of a single variable or in two equations in two unknowns.
5. Test the critical points and endpoints in the domain of the unknown. Use the first and second derivatives to identify and classify the functions critical points.
View the presentations on maximum and minimum applications that include the use of a graphing calculator.
Optimization Problems Practice
1. Find two numbers whose difference is 100 and whose product is a minimum.
Solution: 50 and -50
2. A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
Solution: $163.43
3. Find an equation of a line through the point (3, 5) that cuts off the least area from the first quadrant.
Solution:
4. At which point on the curve does the tangent line have the largest slope?
Solution: (-2,-223) and (2,225)
5. A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height to the nearest tenth of a centimeter of the cup that will use the smallest amount of paper.
Solution: 3.7cm
Optimization Problems: Even More Problems!
Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of optimization.
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