Using automatic_names also makes it so you don't have to use the foo.method(...) notation:

Problem: Create the following expressions: $\sin^5(x)\cos^2(x), \qquad \displaystyle \frac{x^3}{x^3 + 1}, \qquad k\cdot P \cdot \left(1 - \frac{P}{K}\right)$.

Tip: that you must put in an asterisk (*) for multiplication.

Tip: You can edit the HTML between cells by double clicking on it.  You can create new HTML areas by shift-clicking on the blue bar that appears just above a compute cell.

Most standard functions are defined in Sage.  They are named lowercase, e.g.,

sin, cos, tan, sec, csc, cot, sinh, cosh, tanh, sech, csch, coth, log, exp, etc.

Problem: Construct the symbolic expression $\sin(x^{\cos(y)} + \theta) + \coth(2x) + \log(3x)\cdot \exp(y^3)$. 

Making substitutions

Use the subs method to replace any variables by other variables.

Problem: Replace $x$ by $\sin(y)-x$ in the expression $x^3 + x y$.

Expanding Expressions

To expand a symbolic expression with exponents, use the expand method (or function) -- we saw this above:

Problem: Expand the expression $(2\sin(x)  - \cos(y))^5$.

Symbolic Unit Conversions

As of Sage-4.4, Sage has a massive symbolic unit conversion package, which was written by David Ackerman (a UW undergrad) recently, as  a funded student project after he took Math 480 (the Sage class) last year.

SI derived unit of force.
Defined to be kilogram*meter/second^2.

Problem:  Convert 17 meters/second^2 to miles/minute^2.

Problem: Do a simple conversion that might come up in your own research.

Creating Symbolic Functions

To create a symbolic function, use the notation f(x,y) = x^3 + y.  A symbolic function is just like a symbolic expression, except you can call it without having to explicitly use subs or name variables and be sure that the order is what you want.

fast_float: this is critical to know about if you're serious about using numerical Python tools that involve repeated function evaluation.   (See also fast_callable for complex and other inputs.)

Problem: Create the functions $x\mapsto x^3 + 1, \qquad (x, y) \mapsto \sin(x) - \cos(y)/y, \qquad (a,x,\theta)\mapsto a x + \theta^2$.

Problem: Evlauate $\sin (x) - \cos(y)/y$ quickly using fast_float.

2D Plotting

Use the plot command to plot a function of one variable.  TIP: Type plot(<tab key> to find out much more about the plot command.

The plot command has many options:

Note for MATLAB Users: You can also use a full emulation of matlab's plotting system via pylab (which is part of the standard matploblib Python library):

Problem: If you know MATLAB, try testing some simple 2d plotting.

Here's a the same plot, but you can adjust many of the parameters to the plot command interactively.

Problem: Use the above interactive plotter to draw the following plot of $\sin(x^2)$.

You can plot many other things, including polygons, parametric plots, polar plots, implicit plots, etc.:

line, polygon, circle, text, polar_plot, parametric_plot, circle, implicit_plot

You superimpose plots using +.

You can also do 3D plots, including parametric lines, parametric surfaces, implicit 3d plots, etc.

More Symbolic Calculus: Computing Integrals and Derivatives

You can symbolically integrate or differentiate functions, compute limits, Taylor polynomials, etc.

Symbolic Taylor Series

Other Options for Symbolic Calculus

1. Maxima -- a powerful symbolic computer algebra system, which is fully usable from Sage, and included in Sage.  Sometimes you want to do something, e.g., solve some weird differential equation or evaluate a special functions, and you search the web, find how to do it in Maxima, and can then... paste and do the same in Sage via the Maxima interface.

2. SymPy -- a powerful pure Python library for symbolic Calculus, written almost entirely by Physics graduate students.  This can be installed into anything that you can run Python on, and provides a nice range of capabilities.  Sage uses it some, and it is included in Sage.

Maxima