Optimization
Unit 1 Lesson 1: Optimization with Parameters
In this lesson we will review optimization in 2-space and the calculus concepts associated with it.
Learning Objective: After completing this lesson, you will be able to model problems described in context and use calculus concepts to find associated maxima and minima using those models. You will be able to justify your results using calculus and interpret your results in real-world contexts.
We will begin our review with a problem in which most fixed values are designated by letters or symbols. We refer to these values as parameters--they are not variables, but they are represented by letters or symbols.
Let’s start with a short video that shows a basic optimization problem with one parameter before moving on to a more complex example.
Example: Drag Machines (or Vanes)
You are probably familiar with waterwheels and windmills. Each of these is an example of a drag machine, or vane. These machines are used to generate energy using water or wind and have greatly evolved since they were first invented centuries ago. The primary objective when using these machines is to draw the most energy possible from the water or air used.
Click on the video icon to the right to view a video of a small waterwheel that generates a small amount of power.
In this example, we will explore a model that describes the amount of energy such a machine can generate using wind.
Click on the video icon to introduce you to a machine that converts mechanical energy generated by wind to electricity, called a wind turbine:
Palm Springs, CA, is the site of a large center for wind turbines to generate power.
(http://palmsprings.com/services/wind.html) Suppose you are on a tour of the windmills, and you calculate the density, , and velocity, , of the air that passes through them, and the surface area of the vanes is . The force exerted on the vanes’ surfaces is described by the following model:
( )
(
) ,
In this lesson we will review optimization in 2-space and the calculus concepts associated with it.
Learning Objective: After completing this lesson, you will be able to model problems described in context and use calculus concepts to find associated maxima and minima using those models. You will be able to justify your results using calculus and interpret your results in real-world contexts.
We will begin our review with a problem in which most fixed values are designated by letters or symbols. We refer to these values as parameters--they are not variables, but they are represented by letters or symbols.
Let’s start with a short video that shows a basic optimization problem with one parameter before moving on to a more complex example.
Example: Drag Machines (or Vanes)
You are probably familiar with waterwheels and windmills. Each of these is an example of a drag machine, or vane. These machines are used to generate energy using water or wind and have greatly evolved since they were first invented centuries ago. The primary objective when using these machines is to draw the most energy possible from the water or air used.
Click on the video icon to the right to view a video of a small waterwheel that generates a small amount of power.
In this example, we will explore a model that describes the amount of energy such a machine can generate using wind.
Click on the video icon to introduce you to a machine that converts mechanical energy generated by wind to electricity, called a wind turbine:
Palm Springs, CA, is the site of a large center for wind turbines to generate power.
(http://palmsprings.com/services/wind.html) Suppose you are on a tour of the windmills, and you calculate the density, , and velocity, , of the air that passes through them, and the surface area of the vanes is . The force exerted on the vanes’ surfaces is described by the following model:
( )
(
) ,