Additional Resources for Calculus:

 

Software For Doing Math:


(Ordered by ease of use, beginning with the easiest)
link Description
Wolfram|Alpha Wolfram|Alpha: The makers of Mathematica (a very powerful mathematical software package, popular amongst applied mathematicians and engineers) bring us a "google-like" mathematical search engine.

In their own words: "Wolfram|Alpha is the first step in an ambitious, long-term project to make all systematic knowledge immediately computable by anyone. Enter your question or calculation, and Wolfram|Alpha uses its built-in algorithms and a growing collection of data to compute the answer.

Type in your name, it will give you data about how many people in the US share your first name. Type in a function, it will tell you lots of things about it (yes, including it's derivative and indefinite integral). This site only launched 1.5 years ago and I put this link here as yet another resource you may find helpful in learning calculus.

Wolfram|Alpha is also available as a ipod touch/iphone app for $1.99. Of course you could always use it for free in the browser, but the app adds some nice features like a mathematics keyboard.

Here is a short youtube video about how to use Wolfram|Alpha in calculus: http://www.youtube.com/watch?v=vyfhzhfuz58
Geogebra Geogebra is a free software package for visualizing mathematics. It is great for easily creating dynamic interactive plots and geometric constructions.

I'll be using it quite a bit in class, and all interactive applets below were created with Geogebra.

Want to download your own free copy, find it here http://www.geogebra.org.
Desmos Online Graphing Utility Desmos is a free online graphing utility.
Sage SageMath is a very powerful mathematical software package. It is open source and hence free. We'll use it a few times in this course, but I'll provide you with the code and you won't need to download anything. However, if you are familiar with Python and would like to see what professional mathematicians use to do research then play around with SageMath.

Tutorial: For an online tutorial in using SageMath click here. Then click Sage as a Calculator to see basic arithmetic examples, and the left menu now has links to examples from calculus.

Use Sage now: To run SageMath online instantly, try this cell server.

CoCalc: You can access your own creation space by using SageMath online here https://cocalc.com/. Just create a username and password and you're off and running.

Download: If you want to download your own free copy of SageMath to run on your computer then go here www.sagemath.org.
Maple Maple is a very powerful mathematical software package. It is commercial, and does cost money. However, it is installed on most computers on the SFU campus, so if you are an SFU student you do have access to it.

SFU Students will learn Maple in MACM 204 Computing with Calculus (covers differential and integral caculus in one and two variables and differential equations).
Matlab Matlab is a very powerful mathematical software package. It is commercial, and does cost money. However, it is installed on most computers on the SFU campus, so if you are an SFU student you do have access to it.

SFU Students will learn Matlab in MACM 203 Computing with Linear Algebra.
Python Python is a programming language, but it does have libraries that are available to do mathematical computing. It is free and open source.

SFU students have access to python through https://sfu.syzygy.ca/. Login with your SFU ID, and you can use Python in your web browser. You'll also be able to store your notebooks in the cloud.

General public can access to python through https://pims.syzygy.ca/. Login with your google account, and you can use Python in your web browser. You'll also be able to store your notebooks in the cloud.

If you are using Python locally on your computer, you'll want to make sure you have four libraries installed: NumPy, SciPy, matplotlib and SymPy. A quick search will show you how to install these libraries.

For a sample of how Python can be used to do mathematics see this page (still in development).
 

Applets For Exploring Calculus:


Visualizing Functions
link title related section Description
Function Graph Visualizing Functions as Graphs 1.1 A graph of a functions is a visual representation of the pairs (input, output), in the plane. This applet will help you to understand the connection between the graph of a function and a function as and input-output machine.
trig applet Graphs of Sine and Cosine 1.2 An applet illustrating how the graphs of sine and cosine are related to the unit circle.
Transformations Transformations of Functions 1.3 An applet illustrating how transformations affect the graph of a function. Transformations are represented both algebraically and graphically.

Differential Calculus
Tangent to Parabola Construction of the Tangent to a Parabola 2.1 An applet illustrating how the tangent to a parabola is obtained through successive secant lines.
Graphs with Corners Functions with Corners 2.1 Some examples of graphs with corners which I talked about in lecture. The tangent line to a graph does not exist at a corner.
Algebraic Simplification Algebraic Simplification 2.3 When computing the limit of a function at a point it is sometimes necessary to do some algebraic simplification of the function before you can substitute in the value. What we are actually doing is replacing the function with another function which is equal to it but has a larger domain (i.e. the simplified function has the value x=a in its domain). We can think of this as filling in the hole on the original function.
Squeeze Theorem Example Squeeze Theorem Examples 2.3 The Squeeze Theorem is a very useful theorem which allows you to compute limits of some trickier functions. This applet allows you to explore the Squeeze Theorem visually. Here is another example.
Epsilon-Delta Game Epsilon-Delta Definition of Limit 2.4 The epsilon-delta definition of the limit can be difficult to grasp at first. This is an interactive illustration of the epsilon-delta game to help get your head around what is going on.
Derivative of sin The Derivative of Sine 3.3 So you know the derivative of sine is cosine, but do you know why? This applet gives some pretty compelling visual evidence to suggest why. Of course, a rigorous argument would rely on applying the definition of derivative.
Motion of a Particle Motion of a Particle in 1-Dimension 3.7 If a particle has negative acceleration then it must be slowing down, right? Nope. Sorry. Try again. This is an extremely common misconception. This applet is intended to address this misconception and help you understand what the sign of the first and second derivative is telling you. The sign of the first derivative indicates the direction of motion, whereas the sign of the second derivatice indicates the direction of the acceleration. A partical is slowing down if it is accelerating in the opposite direction from which it is moving. This means we also have to know its direction of motion to conclude anything about whether it is speeding up or slowing down.
Rocket Applet Related Rates - Tracking a Rocket 3.9 A visualization of a classic related rates problem - a radar station tracking a launching rocket.
Shadow Applet Related Rates - Man's Shadow 3.9 A visualization of a classic related rates problem - a man walking away from a lamppost.
Lighthouse Applet Related Rates - Lighthouse Beam 3.9 A visualization of a classic related rates problem - a beam of light from a lighthouse is traveling along the shorline.
Mean Value Theorem Mean Value Theorem 4.2 An illustration of the Mean Value Theorem. Rolle's Theorem is included as a special case.
Optimization: Art Gallery Optimization - Art Gallery 4.7 A visualization of a classic optimization problem: Where is the best spot to stand to view a painting?
Optimization Run and Row Optimization - Rowing and Running 4.7 A visualization of a classic optimization problem.
Optimization Soccer Optimization - Soccer 4.7 Investigate the optimum position on the soccer field to score a goal.
Optimization Elvis the Dog Optimization - Elvis the Dog 4.7 Elvis the Calculus Wonderdog!
Newton's Method Newton's Method 4.8 A dynamic illustration of Newton's Method for approximating roots of functions.
Sage Newton's Method Implementation 4.8 Implementation of Newton's Method. Don't bother using your calculator to crunch the numbers, instead use a computer - it is what they are great for. Just change the function and the initial value and compute away!

Here is a pure python version in a jupyter notebook format: (view) (download: .ipynb)

Parametric Curves and Polar Coordinates
Lissajous Curves Circle, Ellipses and Lissajous Parametric Curves Applet 10.1 Some of the most basic parametric curves. Here we see that circles, ellipses and Lissajous curves all arise from sine and cosine through parametrizations.
Parametric Curves Parametric Curves Applet 10.1 This applet is to help you visualize how a parametric curve is constructed from two individual functions: one for the x coordinate and one for the y-coordinate.

(currently this one doesn't work on all mobile devices - it needs java to run)

Here is another applet in which you can plot polar curves [java] [html5].
Polar Curves Polar Curves Plotter 10.3 Just a polar curve grapher. Type in your polar equation and investigate the graph.
Cartesian Curve to Polar Curve Polar Curves and Cartesian Graphs 10.3 An applet showing the connection between the Cartesian graph of r=f(θ) and the graph in polar coordinates.

Integral Calculus
Riemann Sums Riemann Sums Applet 5.1 This applet is to help you visualize Riemann sums. You can change the function, move the sliders to change from left to right Riemann Sums, and increase the number of rectangles. Notice the more rectanges you use, the closer they approximate the actual area under the curve.
Approximate Integration Approximate Integration Applet 7.7 This applet is to help you visualize the techniques we have developed for approximating integrals: left-endpoint, right-endpoint, and midpoint methods, trapezoid rule, and Simpson's rule.
Sage Implementations of lefthand, righthand, and midpoint rules, trapezoid rule, and Simpson's rule. Don't bother using your calculator to crunch the numbers, instead use a computer - it is what they are great at. Just change the function, interval and n, and compute away!

Here is a pure python version in a jupyter notebook format: (view) (download: .ipynb)
Sage Infinite Series: Approximations to famous mathematical constants - by Nils Bruin

Here is a pure python version in a jupyter notebook format: (view) (download: .ipynb)
Sage Tools for visualizing differential equations: Vector Field Plotter - by Nils Bruin

External Links & Guides:
link Description
Stewart's Calculus This is the official webpage of James Stewart's Calculus text (used for Math150/151/152/251). Have a look at the TEC (Tools for Enriching Calculus) link. There are some nice interactive animations.
How to Ace Calculus How to Ace Calculus: The Streetwise Guide is witty little book with good information. Who said reading math had to be dull? This link is to the authors website where they have made much of the information available. You can also get you own copy of the book from places such as Amazon.com.
Calculus.org Calculus.org: Your one-stop-shop for all things calculus. This site contains fully worked out examples, a collection of practice exams from other univeristies, and lots of applets to satisfy your every need. This site is maintained by some well known, and entertaining, mathematical educators. Two of which are the authors of the "How to Ace Calculus" book above. Enjoy!

The first link on the page is to a great collection of practice problems with full solutions.
Letter to a Calculus Student You are about to "hold Infinity in the palm of your hand"! Are you excited? This is a letter to a Calculus student (i.e. you) about the beauty they are about to uncover in Calculus. It was written by Keith Devlin, author of dozens of mathematics books for a general audience.
Polya's Approach to Problem Solving A summary of Polya's 4-step approach to problem solving. A must read for any mathematics student!!! This guide contains a one-page expanded version of the 4-steps.


switch to Applets for Calculus canvas.sfu.ca J. Mulholland