Aaron Schlegel's Notebook of Interesting Things

The Substitution Rule is another technique for integrating complex functions and is the corresponding process of integration as the chain rule is to differentiation. The Substitution Rule is applicable to a wide variety of integrals, but is most performant when the integral in question is similar to forms where the Chain Rule would be applicable. In this post, the Substitution Rule is explored with several examples. Python and SymPy are also used to verify our results.

Antiderivatives, which are also referred to as indefinite integrals or primitive functions, is essentially the opposite of a derivative (hence the name). More formally, an antiderivative \(F\) is a function whose derivative is equivalent to the original function \(f\), or stated more concisely: \(F^\prime(x) = f(x)\). The Fundamental Theorem of Calculus defines the relationship between differential and integral calculus. We will see later that an antiderivative can be thought of as a restatement of an indefinite integral. Therefore, the discussion of antiderivatives provides a nice segue from the differential to integral calculus. The process of finding an antiderivative of a function is known as antidifferentiation and is the reverse of differentiating a function.

Newton's method, also known as Newton-Raphson, is an approach for finding the roots of nonlinear equations and is one of the most common root-finding algorithms due to its relative simplicity and speed. The root of a function is the point at which \(f(x) = 0\). This post explores the how Newton's Method works for finding roots of equations and walks through several examples with SymPy to verify our answers.

An implicit function defines an algebraic relationship between variables. In this post, implicit differentiation is explored with several examples including solutions using Python code.

The chain rule is a powerful and useful derivation technique that allows the derivation of functions that would not be straightforward or possible with the only the previously discussed rules at our disposal. The rule takes advantage of the "compositeness" of a function. In this post, we will explore several examples of the chain rule and will also confirm our results using the SymPy symbolic computation library.

A function limit, roughly speaking, describes the behavior of a function around a specific value. Limits play a role in the definition of the derivative and function continuity and are also used in the convergent sequences. In this post, we will explore the definition of a function limit and some other limit laws using examples with Python.

Implicit differentiation can also be employed to find the derivatives of logarithmic functions, which are of the form \(y = \log_a{x}\). In this post, we explore several derivatives of logarithmic functions and also prove some commonly used derivatives. The symbolic computation library SymPy is also employed to verify our answers.

Several rules exist for finding the derivatives of functions with several components such as \(x \space sin \space x\). With these rules and the chain rule, which will be explored later, any derivative of a function can be found (assuming they exist). There are five rules that help simplify the computation of derivatives, of which each will be explored in turn. We will also take advantage of SymPy to perform symbolic computations to confirm our results.

In this post, we explore the definition of a continuous function and introduce several examples with Python code.

In this example, we walk through a sample use case of extracting data from a database using an API and then structuring that data in a cohesive manner that allows us to create a relational database that we can then query with SQL statements. The database we will create with the extracted data will use Postgresql. The Python libraries that will be used in this example are poetpy, a Python wrapper for the PoetryDB API written by yours truly, pandas for transforming and cleansing the data as needed, and sqlalchemy for handling the SQL side of things.

The poetpy library is a Python wrapper for the PoetryDB API. The library provides a Pythonic interface for interacting with and extracting information from the PoetryDB database. In this introductory example, we will explore some of the basic functionality of the poetpy library for interacting with the PoetryDB database.

The Austin Animal Center provides its animal intake and outcome datasets on Socrata. When an animal is taken into the shelter, it is given a unique identifier that is also used in the outcomes dataset. We have already investigated and performed exploratory data analysis on the Austin Animal Center's intakes and animal outcomes individually and found several interesting facets of information. In this analysis, we merge the intakes and outcomes dataset using pandas to enable us to perform exploratory data analysis on the merged data. With the data merged, we will be able to explore in more depth the transition from intake to outcome.

The Austin Animal Center, the largest no-kill municipal shelter in the United States, makes available its collected data on Austin's Open Data Portal. This data includes both animals incoming into the shelter and the animals' outcome. In this post, we perform some exploratory data analysis on the intakes dataset to see if we can find any noticeable trends or interesting pieces of information of the data. First, we will extract the data from Austin's Data Portal, which is supported by Socrata. We will then perform some data transformation and cleaning steps to get the data ready for analysis.

The city of Seattle makes available its database of pet licenses issued from 2005 to the beginning of 2017 as part of the city's ongoing Open Data Initiative. This post will explore extracting the data from Seattle's Open Data portal using requests, then transform the extracted JSON data into a workable dataset with pandas to analyze and investigate the pet license database.

Following from the previous analyses of the Austin Animal Center's shelter outcomes dataset, we now take what we learned from the exploratory data analysis component of the investigation and build and train a machine learning model for predicting if a cat entering the shelter will be adopted or transferred to a partner facility. Adoptions and transfers make up about 90% of all the outcomes.

The following post walked through how to extract and transform the shelter outcome data to make it tidy and suitable for data analysis. In this post, we perform exploratory data analysis using pandas and seaborn to investigate and visualize the shelter outcomes of cats. The findings that are garnered from the exploratory data analysis step can help tremendously in the model building phase when we need to select the important features of the data.

The consumer complaints database is a collection of complaints received by the Bureau of Consumer Financial Protection related to financial products and services. This post explores creating a database file using SQLite and analyzing the data with Pandas and Plotly.

The Austin Animal Center, the largest no-kill municipal animal shelter in the United States, makes available its shelter animal outcomes dataset as patrt of the City of Austin's Open Data program. This post demonstrates how to extract the data from the City of Austin's Open Data portal using the requests library and convert the resulting JSON to a tabular pandas DataFrame. We will then enrich the data by applying feature engineering to the data to add more information, which should help improve the outcome prediction model.

The Greeks are used as risk measures that represent how sensitive the price of derivatives are to change.

Introduces the call and put option pricing using the Black-Scholes formula and Python implementations.