Sage Examples:

Sage Sandbox:

• Sandbox: Try your own code.
• Linear Algebra Sandbox: This has sample matrix commands, but it is left open so you can try your own code.

Numerical Change of Base:

• Base Change: This demo converts whole numbers between different bases.
• Becimal Calculator: This demo does rational number calculations in different bases.

Transcendental Stuff:

• Patterned transcedental numbers: This demo calculates fragments of decimal expansions of patterned transcendental numbers (see Patterned Transcendental Numbers for more details).

Sage Demos for Calculus:

• Limits: This demo attempts to compute left, right, and double sided limits while providing numerical support and a plot. [Limits sandbox]
  Related Desmos Definition of Limit Demo
  
  
• Average Rates of Change: This demo compute average rates of change for a given function (defined by formula or table). The demo also provides a plot to help visualize what is being computed.
  
  
• Definition of the Derivative: This demo plots secant lines and tangent lines. It also computes difference quotients and derivatives. This allows the user to explore the limit definition of the derivative. [Derivatives Sandbox]
  Related Desmos Definition of the Derivative Demo
  
  
• Tangent Lines: This demo plots a function along with a tangent line based at a point determined by a slide bar. Derivative information and the equation of the tangent line are also displayed. If a box is checked, the quadratic approximation (i.e. second order Taylor polynomial) is also computed and displayed.
  Related Desmos Tangent Line Demo
  
  
• Derivatives Graphs: This demo plots a function and its first and second derivatives along with points based at $x=a$ on these graphs (this point is determined by a slider). Using the derivative values, it is determined whether the function is increasing, decreasing, or has a critical point and whether it is concave up, concave down, or possibly has an inflection point.
  Related Desmos Graphs of Derivatives Demo
  
  
• Derivatives Practice: This is a Sage enabled version of a derivatives practice worksheet. The demo computes and prints the derivatives of a given function up to a specified order.
  
  
• Taylor Polynomials: This demo computes and graphs Taylor polynomials. There is an option whether to show a single polynomial, all polynomials up to a specified order, or animate a sequence of polynomials.
  
  
• Newton's Method: This demo computes roots using Newton's method. It also provides an animated visualization of the method by providing graphs of tangents.
  
  
• Critical Points: This demo computes critical points and solutions of $f''(x)=0$. These points are plotted and classified (i.e. local min, local max, neither and inflection point or not).
  
  
• Optimization: This demo computes the minimum and maximum value of a function $f(x)$ when $a \leq x \leq b$. The demo also includes a plot of the function and highlights the critical points and end points.
  
  
• Area Approximation: This demo computes left, right, midpoint, trapezoid, and Simpson's rule approximations of $\int_a^b f(x)\,dx$. The demo also produces a plot to show what is being computed.
  
  Area Approximation II: The demo computes left, right, and trapezoid rule approximations of $\int_a^b f(x)\,dx$. The demo also produces a plot to show what is being computed. Note: This demo allows for general partitions (not just equally spaced) and it also allows one to specify a function using a table of values.
  
  

Robot Arm:

• Robot Arm: This demo plots a robot arm with a claw whose configuration space is 6-dimensional. [Robot Arm with open code window, robot_arm.sage, and Robot Arm in Maple (.mw)]
  
  

The Euclidean Algorithm and Some Algebra Demos:

• Eucldiean algorithm for integers: The demo runs the extended Euclidean algorithm on a pair of integers computing their greatest common divisor.
  
  
• Eucldiean algorithm for polynomials: This demo runs the extended Euclidean algorithm on a pair of polynomials in $\mathbb{Q}[x]$ (rational coefficients) or $\mathbb{Z}_n[x]$ (coefficients modulo $n$).
  
  
• Descartes' rule of signs: This demo uses Descartes' method to give an upper bound on the number of positive and negative real roots of a polynomial by merely looking at the sign pattern of the polynomial's coefficients.
  
  
• Budan-Fourier method: This demo uses the Budan-Fourier theorem to give an upper bound on the number of real roots lying in a speicified interval.
  
  
• Sturm's method: This demo uses Sturm's method to give an exact count of the number of distinct real roots lying in a specified interval.
  
  
• (Advanced) Stum's method: This demo iterates Sturm's method to give the exact number of real roots in a specified interval counting multiplicities. The demo also allows the user to produce a table of sign patterns for the usual Sturm's method.
  
  
• Horner's method: This demo performs polynomial division with remainder by computing a minimal table of values.
  
  

Lie Algebras:

• Root Systems (BETA) This demo displays various data associated with a specified simple Lie algebra. Given a highest weight, the weights of the corresponding irreducible representation are listed. For systems of ranks 1, 2, or 3, a plot of the weights (and weight lattice) is displayed.
  
  Note: I believe this demo works correctly - but - it has basically no documentation.