Calculus with Julia

This is a set of notes for learning calculus. Since the mid 90s there has been a push to teach calculus using many different points of view. The Harvard style rule of four says that as much as possible the conversation should include a graphical, numerical, algebraic, and verbal component. These notes use the programming language Julia to illustrate the graphical, numerical, and, at times, the algebraic aspects of calculus.

There are many examples of integrating a computer algebra system (such as Mathematica, Maple, or Sage) into the calculus conversation. Computer algebra systems can be magical. The popular WolframAlpha website calls the full power of Mathematica while allowing an informal syntax that is flexible enough to be used as a backend for Apple's Siri feature. ("Siri what is the graph of x squared minus 4?") For learning purposes, computer algebra systems model very well the algebraic/symbolic treatment of the material while providing means to illustrate the numeric aspects. Theses notes are a bit different in that Julia is primarily used for the numeric style of computing and the algebraic/symbolic treatment is added on. Doing the symbolic treatment by hand can be very beneficial while learning, and computer algebra systems make those exercises seem kind of pointless, as the finished product can be produced much easier.

Our real goal is to get at the concepts using technology as much as possible without getting bogged down in the mechanics of the computer language. We feel Julia has a very natural syntax that makes the initial start up not so much more difficult than using a calculator. The notes restrict themselves to a reduced set of computational concepts. This set is sufficient for working the problems in mathematics, but do not cover thoroughly many aspects of programming. (Those who are interested can go off on their own and Julia provides a rich opportunity to do so.) Within this restricted set, are operators that make many of the computations of calculus reduce to a function call of the form action(function, arguments...). With a small collection of actions that can be composed, many of the problems associated with introductory calculus can be attacked.

These notes are presented in pages covering a fairly focused concept, in a spirit similar to a section of a book. Just like a book, there are try-it-yourself questions at the end of each page. All have self-graded answers.

Getting started with Julia

Before beginning, we need to get started with Julia. This is akin to going out and buying a calculator, though it won't take as long.

Precalculus

Many of the necessary computational skills needed for employing Julia successfully to assist in learning calculus are in direct analogy to concepts of mathematics that are first introduced in precalculus or prior. This precalculus review, covers some of the basic materials mathematically, but more importantly illustrates the key computional mechanics we will use throughout. A quick rundown of the julia concepts alone is here.

Number systems

Taking for granted a familiarity with basic calculators, we show in these two sections how Julia implements the functionality of a calculator in a manner not so different.

Calculators really only use one type of number – floating point numbers. Floating point numbers are a model for the real numbers. However, there are many different sets of numbers in mathematics. Common ones include the integers, rational numbers, real numbers, and complex numbers. As well, we discuss logical values and vectors of numbers. Though integers are rational numbers, rational numbers are real numbers, and real numbers may be viewed as complex numbers, mathematically, these distinctions serve a purpose. Julia also makes these distinctions and more.

An arithmetic progression is a sequence of the form $a, a+h, a+2h, \dots, a+kh$. For example $3, 10, 17, 24, .., 52$. They prove very useful in describing collections of numbers. We introduce the range operator that models these within Julia and comprehensions that allow one to easily modify the simple sequences.

Functions

The use of functions within calculus is widespread. This section shows how the basic usage within Julia follows very closely to common mathematical usage. It also shows that the abstract concept of a function is quite valuable.

A graphing calculator makes it very easy to produce a graph. Within Julia, the add-on Gadfly package, used through the Plots package,k makes it even easier.

Polynomials

Polynomials play an important role in calculus. They give a family of functions for which the basic operations are well understood. In addition, they can be seen to provide approximations to functions. This section discusses polynomials and introduces the add-on package SymPy for manipulating expressions in Julia symbolically. (This package uses the SymPy library from Python.)

The roots of a univariate polynomial are the values of $x$ for which $p(x)=0$. Roots are related to its factors. In calculus, the zeros of a derived function are used to infer properties of a function. This section shows some tools in SymPy to find factors and roots, when they are available, and introduces the Roots package for estimating roots numerically.

A rational expression is the ratio of two polynomial expressions. This section covers some additional details that arise when graphing such expressions.

Trigonometric functions

Trigonometric functions are used to describe triangles, circles and oscillatory behaviours. This section provide a brief review.

Limits and Continuity

The notion of a limit is at the heart of the two main operations of calculus, differentiation and integration.

Continuous functions are at the center of any discussion of calculus concepts. These sections define them and illustrate a few implications for continuous functions.

Derivatives

The derivative of a function is a derived function that for each $x$ yields the slope of the tangent line of the graph of $f$ at $(x,f(x))$.

The derivative of a function has certain features. These next sections explore one of the first uses of the derivative – using its zeros to characterize the original function.

The tangent line to the graph of a function at a point has slope given through the derivative. That the tangent line is the best linear approximation to the curve yields some insight to the curve through knowledge of just the tangent lines.

The derivative finds use outside of the traditional way of specifying a function or relationship. These two sections look at some different cases.

Integration

The integral is initially defined in terms of an associated area and then generalized. The Fundamental Theorem of Calculus allows this area to be computed easily through a related function and specifies the relationship between the integral and the derivative.

Integration is not algorithmic, but rather problems can involve an array of techniques. Many of these are implemented in SymPy. Theses sections introduce the main techniques that find widespread usage.

Applications

Various applications of the integral are presented. The first two sections continue with the idea that an integral is related to area. From there, it is seen that volumes and arc-lengths may be expressed in terms of related integrals.

Ordinary differential equations are an application of integration and the fundamental theorem of calculus.

Two-dimensional curves

In addition to curves generated by graphs of functions, there are other ways to describe curves in two dimensions. One involves parameterizing both the $x$ and $y$ coordinates by a third variable, such as time. The other, polar curves, describes position in terms of a angle from the positive $x$ axis and a radial distance.

Bibliography