Aaron Schlegel's Notebook of Interesting Things

Integration by parts is another technique for simplifying integrands. As we saw in previous posts, each differentiation rule has a corresponding integration rule. In the case of integration by parts, the corresponding differentiation rule is the Product Rule. This post will introduce the integration by parts formula as well as several worked-through examples.

L'Hospital's Rule allows us to simplify the evaluation of limits that involve indeterminate forms. An indeterminate form is defined as a limit that does not give enough information to determine the original limit. In this post, we explore several examples of indeterminate forms and how to calculate their limits using L'Hospital's Rule. We also leverage Python and SymPy to verify our answers.

The Fundamental Theorem of Calculus is a theorem that connects the two branches of calculus, differential and integral, into a single framework. We saw the computation of antiderivatives previously is the same process as integration; thus we know that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus formalizes this connection. The theorem is given in two parts, which we will explore in turn along with Python examples to verify our results.

As we noted in the previous sections on the Fundamental Theorem of Calculus and Antiderivatives, indefinite integrals are also called antiderivatives and are the same process. Indefinite integrals are expressed without upper and lower limits on the integrand, the notation \(\int f(x)\) is used to denote the function as an antiderivative of \(F\). Therefore, \(\int f(x) \space dx = F^\prime(x)\).

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.