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Optimization and Differentiation

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  • 0:06 Intro to Optimization
  • 0:44 Optimize the Problem
  • 2:07 Five Steps to Optimization
  • 4:58 Lesson Summary
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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.

Introduction to Optimization

Graph showing relationship between hours of sleep and test scores
Optimization Sleep Graph

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.

Optimize the Problem

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.

At this maximum point, the derivative of the test score as a function of sleep is equal to zero
Optimizing the Sleep Problem Graph

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.

Five Steps to Optimization

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.

Graph for the cafe optimization problem
Optimizing the cafe problem graph

Step 1: Visualize It

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.

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