For a more thorough introduction, visit Calculus with Julia.

This is a collection of notes for exploring calculus concepts with the `Julia`

programming language. Such an approach is used in MTH 229 at the College of Staten Island.

These notes are broken into different sections, where most all sections have some self-grading questions at the end that allow you to test your knowledge of that material. The code should be copy-and-pasteable into a `julia`

session. The code output is similar to what would be shown if evaluated in an `IJulia`

cell, our recommended interface while learning `julia`

.

The notes mostly follow topics of a standard first-semester calculus course after some background material is presented for learning `julia`

within a mathematical framework.

Basics of types, order of operations, assignment and variables.

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Shows how to define and call a function. Technical parts include ternary operator, multiple arguments, and return values (tuples).

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This demonstrates the use of the `Plots`

package for plotting. This package provides a very simple `plot`

interface for graphing one or more functions.

As well, a discussion about arrays and mapping a function over an array is given. This will be useful later on with limits, ...

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Finding zeros for polynomials, graphically finding zeros, and using the bisection method.

The add-on `Roots`

package provides some convenient functionality.

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Basics of limits.

Discussion on floating point representation and potential issues (subtracting like-sized objects!)

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Explore forward difference and central difference with a bit on error analysis.

We end with a brief discussion on automatic differentiation, as implemented in the `ForwardDiff`

add-on package via the `Roots`

package.

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A look at the relationship between a function and its first and second derivatives.

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Basics of Newton's method with a copy-and-paste function to do the work after the student explores a bit.

Discusses iterative algorithms, approximation, some analysis.

The `fzero`

function of the `Roots`

package is discussed.

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A project on minimization and maximization. Some standard applied problems are presented.

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Basics of integration with applications including rectangle, trapezoid, Simpson's, and the `quaggk`

function. Applications to volumes of solids of revolution.

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Discusses how to do some symbolic math in `julia`

through the `SymPy`

package.

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- Applications of the integral: area between two curves, volume of solids of revolution, other volumes

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- Techniques of integration: substitution, integration by parts, partial fractions

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- Taylor polynomials

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- Parametric equations and polar coordinates

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- Vectors and vector-valued functions, $f: R -> R^n$

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- Functions of several variables, $f:R^n -> R$.

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- Double and triple integration.

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`Julia`

makes an excellent choice for this material as its syntax is very similar to standard mathematical syntax. The ability to define mathematical functions using the familiar `f(x) = ...`

notation makes getting started really easy. Further, the fact that functions are *first-class objects* means that it is possible to create *higher-order* julia functions that mirror the standard operators of calculus. The following pattern is used throughout:

action(function_object, args...)

For example, the notes use:

`plot(f, a, b)`

to plot`f`

over`[a,b]`

(from`Gadfly`

)`plot([f,g], a, b)`

to plot both`f`

and`g`

over the interval`[a,b]`

`roots(f)`

to find the roots of a polynomial function,`f`

(from`Polynomials`

)`fzeros(f)`

to find the real roots of a polynomial function`f`

(from`Roots`

)`fzero(f, [a,b])`

to find a root inside the bracketing interval`[a,b]`

(from`Roots`

)`limit(f, c)`

to find the limit of`f`

at`c`

(from`SymPy`

)`D(f)`

to return a function that computes the derivative of`f`

(from the`Roots`

package)`fzero(f, a)`

or`[fzero(f, x) for x in [x1,x2, ...]]`

to find root(s) of`f`

starting at`a`

or each of the x's`quadgk(f, a, b)`

to find the numeric integral of`f`

over`(a,b)`

(from base`julia`

)`integrate(f)`

to find the symbolic integral of`f`

(from the`SymPy`

package)`integrate(f, a, b)`

to find the definite integral over`[a,b]`

symbolically

With just this basic set of actions, akin to buttons on the calculator, a rich variety of problems can be addressed.

`Julia`

is a young language, with the bulk of its development being done since its initial announcement. It has relatively few online resources. Some are compiled here. Many of these are linked to from a julia web brain.

- The Julia manual provides a comprehensive overview

- some blog posts are collected here.

- At forio.com a tutorial is provided here.

Before starting out with `Julia`

it must be available.

`julia`

In order to get started with `Julia`

it needs to be installed. If this is not done already, you have a bit of work to do to get `julia`

and the notebook interface provided by `IJulia`

.

First to install `julia`

you can download a copy or install it from source. Likely a download is easiest. Official releases are available from julialang.org but it is best to download a cutting-edge release from status.julialang.org. Installation is hopefully similar to what you do for other software on your system.

`julia`

Starting `julia`

varies amongst the different operating systems. All have a *console* where commands are typed for `julia`

to interpret and execute. This is known as the *command line* and though a long familiar means of interacting with computers, it is generally not familiar to the average student. We will need to learn to like the command line. Once done, you may think it is great, but it can a bit frustrating getting to that attitude.

Here is what the command line looks like on startup from a mac book pro within the terminal:

_ _ _ _(_)_ | A fresh approach to technical computing (_) | (_) (_) | Documentation: http://docs.julialang.org _ _ _| |_ __ _ | Type "?help" for help. | | | | | | |/ _` | | | | |_| | | | (_| | | Version 0.4.1 (2015-11-08 10:33 UTC) _/ |\__'_|_|_|\__'_| | Official http://julialang.org/ release |__/ | x86_64-apple-darwin13.4.0 julia>

The command line is the last line: a prompt beginning with `julia>`

. Here is where you type an expression and then the *enter* key to ask `julia`

to evaluate it.

A simple command is then typed into the computer followed by the *enter* key. This is then sent to `julia`

's interpreter and an answer returned:

2 + 2

4

If you get `4`

, you are able to use `julia`

.

The command line is not the most comfortable learning experience for `julia`

, rather it is suggested that the `IJulia`

notebook interface be used. In the `IJulia`

notebook, the command line is replaced by a cell where commands can be entered and executed in batches. The editing of commands is much easier and some features for integrated help are available.

The above graphic is from the main web page for `julia`

(julialang.com) and shows the `IJulia`

notebook with some graphics provided by the `Gadfly`

package.

Using `IJulia`

will require a few additional installation steps:

- download
`anaconda`

(https://store.continuum.io/cshop/anaconda/). It is big, but free. Install it, then within a terminal (or Windows' command propmpt) enter these commands:

conda update conda conda update ipython

- Start
`julia`

then enter these commands to install the packages we use:

Pkg.update() Pkg.add("IJulia") Pkg.add("Gadfly") Pkg.add("Roots")

The above commands form the basics of `julia`

's package system. Like most computer languages, `julia`

can be extended by user-contributed packages. The complete list of available packages is kept on the computer you are using `julia`

at. This list is updated by the command `Pkg.update()`

. New packages are made available for use by installing or `add`

ing them to your system via `Pkg.add`

. Adding packages will automatically install any dependent packages. As well, external libraries *should* also be installed for you. This magic attempts to automatically identify what your computer system needs and acts accordingly.

The above commands need only be done when new packages are being installed. However, each time you wish to actual **use** an external package in a session, it must be added. This is done with the `using`

command, as `using Plots`

Afterwards those commands are successful, the following command will start the notebook interface:

run(`ipython notebook --profile=julia`)

After all the installation, you can start the `IJulia`

interface by simply starting `julia`

, then issuing the above command *or* you can run just the command `ipython notebook --profile=julia`

from the command line.

For now, you can use `Julia`

online through juliabox.org.

`Julia`

can be extended through external packages. Although a relatively young language, there are already over 300 add-on packages readily available for Julia through its package manager.

For example, there are numerous packages that provide means to draw graphs. To list a few: `Gadfly`

, `Winston`

, `PyPlot`

, `Gaston`

, `ASCIIPlots`

, `Plotly`

, `GoogleCharts`

, ... The first few are the main ones. In these notes we use `Plots`

, an interface to many of the above.

In the `julia`

world, a package author may publish his or her package so that it is easy for an end user to use and install. For the end user there are just a handful of important commands to install a package:

- Call
`Pkg.udpate()`

to update the currently installed external packages and to update the list of available packages to install. Though this command can be a bit slow, it is a good idea to run it periodically.

- To add a new package, call
`Pkg.add("package_name")`

, where you have to put the appropriate package name in. For example, the command`Pkg.add("Plots")`

will install the`Plots`

package. In the process, any external dependencies will be resolved. These include installing any packages that the one you want depends on and in some cases, additional software.

- There are other useful commands, but those two are basically it:
`Pkg.update()`

to update and`Pkg.add()`

to add a new package.

External packages must be loaded into a session. This need only be done once. The easiest way is to use the keyword `using`

, a in `using Plots`

. This must be done *before* you try to use any functionality related to the package. For interactive use, it is a good idea to just pull in familiar packages at the outset.

This has some cost, as some packages are slow to load.

So, to make a plot using `Plots`

, the sequence might go like:

using Plots gadfly() # use Gadfly backend f(x) = x^2 - 2x plot(f, -2, 1)

The manual has some more information.