Optimization problems are an essential part of calculus, as they involve finding the maximum or minimum value of a function within certain constraints. These problems often arise in real-world applications where resources are limited, and we need to make the best possible use of them. In this post, we’ll explore what an optimization problem is in calculus and discuss why we should care about them when making a daily fantasy baseball lineup on FanDuel.

## What is an Optimization Problem?

In a typical optimization problem, we are given two equations: one that represents the restriction or constraint and another that we aim to maximize or minimize.

For instance, imagine you have a limited amount of material to build a fence around a rectangular garden. In this case, the restriction equation represents the perimeter of the fence, while the equation to maximize or minimize represents the area of the garden. Solving an optimization problem involves finding the dimensions of the garden that maximize its area while staying within the constraint of the available fencing material.

If you want to read more about how to solve an optimization problem that you would come across in calculus, you can see an example worked out here.

## Why Does it Matter For DFS?

Now, let’s relate this concept to daily fantasy baseball on FanDuel. When creating a lineup, you are given a fixed budget to spend on players, which serves as the constraint. In FanDuel, it’s $35,000 that you have available for your entire lineup. Each player has an associated cost, or salary, and an expected number of fantasy points they will score. The goal is to maximize the total fantasy points of your lineup while staying within the budget constraint.

There are a few additional constraints that FanDuel imposes on your lineup as well in a full roster contest. You need to make sure to have exactly one player in each position slot, with first baseman and catcher consolidated into one slot, and a utility player that can be anything besides a pitcher. You can also only have a maximum of 4 players from the same team.

Clearly there are additional constraints in DFS beyond the constraints of a calculus problem which makes this more complex, but the strategy for solving this optimization problem is the same.

First, we identify the restriction equation, which in this case is the budget constraint combined with the positional and team stacking requirements. Next, we formulate the equation representing the total fantasy points of the lineup, which we want to maximize. Finally, we use calculus techniques to determine the optimal lineup that maximizes fantasy points while adhering to the constraints.

## Calculus Can Be Used in the Real World

This real-world application of optimization problems in calculus highlights my mission statement quite well: “These posts should inspire college algebra and calculus students to learn the basics taught in their required coursework, so that they can build on it with interesting, real-world applications like the ones discussed in these posts.” By connecting calculus concepts to practical scenarios such as daily fantasy baseball, I want to motivate you to master the fundamentals and encourage you to explore the myriad applications of calculus in the world around them.

You just have to get a little creative and play around with the math from time to time.

Understanding optimization problems in calculus opens up a world of opportunities for applying mathematical techniques to real-life situations. Whether it’s maximizing the area of a garden or assembling the perfect fantasy baseball lineup, the principles of optimization can help you make the most of limited resources and achieve the best possible results.

## What’s Next?

In the next post, I’m going to get into how to solve optimization problems a bit more. I’ll show you the steps of a simple example, then we’ll start talking about how to set up a FanDuel lineup as an optimization problem. We’ll approach it more from the math setup perspective, as opposed to the general thought behind it, like in this post.

If you want to play along and enter some FanDuel lineups of your own, you can use my FanDuel referral link here to get a deposit bonus. You should just need to deposit at least $15 within 30 days of signing up, and you’ll get a $15 bonus added to your account if you use that link.

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