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Math 104: Calculus14 chapters | 116 lessons | 11 flashcard sets

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Lesson Transcript

Instructor:
*Heather Higinbotham*

In this lesson, you can learn what optimization means from a mathematical standpoint. Using the techniques taught in this lesson, you can use the five steps to optimization to figure out practical things, like how much sleep you need to get before an exam.

Optimization problems are absolutely everywhere. For example, in college I learned that the number of hours I sleep directly related to a score that I would get on a test the next day. Here's kind of what it would look like. If I got no sleep, I was an absolute zombie. My test score was not great. Then, at about four hours, I got enough sleep, or so I thought. Eight hours was a good amount of sleep, and if I slept too long, I just didn't study enough, so my test score dropped off if I slept more than nine hours or so.

My goal was always to **optimize** this problem; to find the number of hours I needed to sleep to get the highest exam score. If I look at my hours of sleep and my test score, I'm trying to find the number of hours that will give me the highest test score. Well, that's over here at eight. Why? Because at this point, my test score is a maximum, which means that if I could take the derivative of this function, of my test score as a function of sleep, I would find that the derivative at this point is actually equal to zero.

Okay. What's another optimization problem? Say you have a café, and you're trying to minimize the total cost depending on the number of tables that you put up in your café. So say your graph looks something like this. You want to find the number of tables that I need in my café to minimize the overhead costs. So, again, I'm going to look at this minimum of this graph. If I can take the derivative of total cost as a function of the number of tables in the café, I would find that at this minimum the derivative is zero. That's the number of tables that I need in my café and the total cost.

So, in general, **optimization problems** are everywhere in the world. But you really want to follow a five-step approach to solving optimization problems. First, you want to visualize the problem. Make sure you have a good understanding of what the problem actually is. Then you want to define it. Just because you can visualize it doesn't mean you know what the actual problem is. I can see very well that the economy is not doing so hot these days, but that doesn't mean I can define exactly what the problem is. The third step is writing an equation. If you don't have an equation, then this is finding the graph. The fourth step is to find the minimum or maximum of this graph or equation. So if you're on a graph, you can actually look for the minimum or maximum. But if you're not, you're going to want to find the derivative and see where that equals zero and determine if that's a minimum or maximum. You will also want to consider the end points when finding the minimum or maximum. The fifth step is just answering the question. A lot of people can find the minimum or maximum, but you need to make sure that you answer the question you've defined if you have an optimization problem.

For example, let's take my exam grade as a function of the number of hours that I sleep. Then let's visualize it. Okay. So here's me sleeping - drool coming out of my mouth - and here's me studying. It's kind of a visualization. I could probably put my father in here looking very sternly at me if I don't score well or maybe a report card.

Two: I need to define the problem. Well, the problem here is optimizing the number of hours that I sleep with the number of hours that I study. I need to know how many hours to sleep to get the highest test score. That is my problem.

Then I need to write an equation. Well, we don't have an exact equation for this, but we do have a graph. So here's my graph: test score as a function of the hours that I sleep.

Find the minimum or maximum. Well, if I knew the actual function here, I could take the derivative and see where that's zero and determine if that's a minimum or maximum. But I know from this graph that my maximum value is right here at eight hours.

Now I need to answer the question. Okay. Eight hours of sleep will maximize my test score.

Let's take a quick review of **optimization problems**. Optimization problems are all over the physical world. You see them in the economy. You see them in cost analysis of planes. You see them in your own world for how much sleep you should get, or how much money you should save, or how much paint you should use. They're everywhere. To solve them, you want to follow a 5-step process:

- Visualize it
- Define it
- Write an equation for it
- Find the minimum or maximum
- Answer the question

If you do those, you can solve almost any optimization problem given to you.

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Math 104: Calculus14 chapters | 116 lessons | 11 flashcard sets

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