Desmos Demonstrations

This is a collection of some of the novel demonstrations I’ve created in the Desmos Graphing Calculator for college algebra, trigonometry, single-variable calculus, multivariable calculus, etc. Feel free to use these or build on them. Let me know if you expand anything here into something strictly cooler.

Slope Field of a Differential Equation, and a Solution via Euler’s Method

Draws the slope (direction) field for the given differential equation \(y' = f(x,y)\). The movable black point sets the initial condition of an approximated particular solution drawn with Euler’s method. The variable \(V_{\text{iewport}}\) should be fixed, equal to the min/max values of the \(x\)-/\(y\)-axis, and the variable \(h\) controls the space between individual line segments of the slope field.

Polynomial Regression Modelling the Stock Price of Twitter

Plots Twitter (TWTR) closing stock prices each week of 2022 through October 27th, when it was delisted from the NYSE after being bought by Elon Musk, and graphs various even-degree polynomial models for this data computed via Desmos’s built-in regression functionality. It appears Desmos hits a computational limit computing the degree-\(16\) polynomial of best fit.

Graphs of Functions in Polar and Rectangular Coordinates

The function \(f\) is geometrically interpreted as a curve in the plane in two ways: first as its graph \(y=f(x)\) in rectangular (Cartesian) coordinates as the locus of points \(\big(x, f(x)\big)\), and second as its graph \(r=f(\theta)\) in polar coordinates as the locus of (rectangular) points \(\big(r\cos(\theta), r\sin(\theta)\big)\). A bubble traces out corresponding points on these two graphs between \(\theta_{\text{min}}\) and \(\theta_{\text{max}}\), shading in the area “under” the polar graph, green for positive area and red for negative area.

Interactive Graphs of Sine, Cosine, and Tangent Based on the “Unit Circle”

The graphs are traced out as \(t\) moves around the circle’s perimeter.

Demonstration of the Epsilon-Delta Definition of a Limit

Given a function \(f\) continuous at \(c\), this plot illustrates the shrinking \(\varepsilon\)-neighborhood around the limit and \(\delta\)-neighborhood around \(c\) that we must consider when rigorously proving a limit’s value.

Demonstration of Newton’s Method for Approximating a Zero of a Function

Given a differentiable function \(f\) and an initial “guess” \(c_0\) for a root of \(f,\) this plot illustrates seven iterations of Newton’s method of approximating that root of the function \(f.\)

The Nth Taylor Polynomial of a Sufficiently Smooth Function

Graphs the \(N\)th Taylor polynomial approximating a smooth function \(f\) centered at the movable point \(\big(a,f(a)\big)\). The maximum value of \(N\) in this demo is \(16\), which can be expanded simply by expanding the Manual List of Derivatives. Note: Desmos will throw the classic “Definitions are nested too deeply” error for some definitions of \(f\), which can be fixed by deleting some entries, about five, from the end of the Manual List of Derivatives. Additionally there are a couple of explicit examples of functions and their Taylor series included.

The Area Under a Curve Approximated as the Sum of Rectangular Areas

Suppose a Maserati is speeding down a straight highway in Nevada. For a twelve second window of time it has a velocity given by the function \(f(t)=120+2t^2-\frac{1}{12}t^3\). How much distance does the Maserati cover in those twelve seconds? My students, prior to learning integral calculus, will approximate answers to this question a couple different ways. This demonstrates that, in the limit, all those approximations approach the same true answer.