The applets on this page are local implementations of an applet written at Brown University for the calculus class of Tom Banchoff. (More precisely the links are to web pages that contain parameter tags that call that applet in a variety of configurations useful for projects and demonstrations.) The applet is used with permission. The applet was designed for use with multivariable calculus. A help page is available for these applets.

Graphers

Continuity

Differentiation

Integration

Sequences and Series

Vectors

Continuity in several variables

Differentiation of functions of several variables

Integration in vector fields

The Visual Composition of Functions Applet looks at composition of functions by using three variables, with y=g(x) and z=f(y). The projection of the curve obtained from the intersection of the two surfaces is then either the graph of f, g, or fg, depending on the plane it is projected onto.

The Parameterized Curve Applet is set up to look at a parameterized curve in x-y as a space curve in x-y-t space. Rotating the axes lets you see the parameterized curve as well as the x(t) and y(t) curves in the x-t and y-t planes respectively.

The delta-epsilon definition of continuity can be explored with the single variable continuity
applet.
Setting epsilon to a positive value gives horizontal bars.
You want to find a positive delta small enough that the curve is
trapped by the bars.

The secant and tangent line applet shows how a sequence of tangent lines converges to the line tangent to a curve.

The Linear
Approximation applet explores the issue of the region where the
tangent line can be
used as a good approximation of a function.

A visualization of the chain rule starts with composition of functions and understanding that zooming in far enough will make differentiable functions look linear. Then we can verify that the slopes of the related linear functions are relates as products according to the normal chain rule.

Single variable differentiability can be visualized with a
delta-epsilon definition that is very similar to the definition used
for continuity. Setting a positive value for epsilon gives a
range of slopes, and the user must set delta small enough to trap the
curve in the cone defined.

One of the standard applications of the definite integral is to find
the Area
Between Two Curves. This applet helps develop the visual
intuition and check limits with the
curves either to the top and bottom or to the left and right of the
region.

A second application is to find the length of a curve with the curve defined as
the graph of a function.

A third application of the integral is to find the volume of a solid
for special cases. One special case is a solid of
revolution, finding the volume using either the shell method or the
disk method. A second case is a solid defined by a base and a given
shape for cross sections. We find volume by using geometry to find
the area function for the cross section and integrating the area
function to find volume.

A theme of single variable calculus is how the Taylor polynomials
can be used to approximate functions. A Taylor
Polynomial explores the same themes.

Theses applets are designed to be used for visualization exercises in multivariable calculus.

The Cross Section applet looks at how the graph of a 2 variable function is built up from the cross graphs of the cross sections, which are each graphs of single variable functions.

The Level Curve applet builds up the graph from its level sets. It connects the contours of a three dimensional surface with the contour graph obtained by looking down the z-axis. (Currently, this one has bugs with Safari.)

The DirectionalSlice applet lets you take a cross section of a surface that includes a given line segment on the x-y plane.

The 3D Grapher Applet page collects 3 applets for graphing in 3-space. One can either use the Cartesian Grapher to graph z as a function of x and y, or use the Cylindrical Grapher to graph r as a function of theta and z, or use the Spherical Grapher to graph r as a function of theta and phi.

The Parameterized Curve Applet is set up to look at a parameterized curve in x-y as a space curve in x-y-t space. Rotating the axes lets you see the parameterized curve as well as the x(t) and y(t) curves in the x-t and y-t planes respectively. The Parameterized Surface Applet lets you see the graph of a parameterized surface. As we did with the cross section applet above, we can also see a wire frame constructed from the graphs of lines in the domain.

The Adding Vectors in
3D applet gives a visualization of addition of 2 vectors in 3
dimensions.

The Adding 4 Vectors
in 3D applet allows 4 vectors in 3D to be added together.

The Linear Combinations
and Subspace applet shows the linear
combinations of two vectors, U and V, in R^{3}, and the
subspace spanned by those vectors.

Two similar applets are available to look at cross products.
They differ in the mechanism used to control the vectors.
The first Cross
Product
applet gives a visual approach to cross products of vectors in 3-space
with the coordinates of the two vectors controlled by sliders.
The second Cross Product
applet gives a visual approach to cross products of vectors in 3-space
with the coordinates of the two vectors controlled by text boxes.

The Vector Field
applet allows simple graphing of vector fields, in either 2 or 3
variables.

The applet Tangent Planes, connects the tangent lines of the x and y cross sections at a point to the tangent plane to the surface at that point.

The Differentiability applet explores the definition of differentiability as an extension of the definition of continuity. This is a generalization of an applet that looks at single variable differentiability.

The Critical Point Applet lets the user explore how to classify critical points of functions of two variables.

The Polar Functions Applet looks at a collection of functions that are useful in seeing what can go right or wrong when thinking about differentiability of functions in two variables.

The Multivariable
Linear
Approximation explores the issue of the region where the
tangent plane can be
used as a good approximation of a function.

The Multivariate
Chain Rule applet looks at the chain rule in the case of
composition of
functions from R to R^{2} to R.

The Line Integral Applet explores a visualization of the line integral of a vector field over a parameterized curve.

The Flux Integral Applet explores a visualization of the flux integral of a vector field over a parameterized surface.

Return to the Saint Louis University Department of Mathematics and Computer Science home page

Return to the SLU Math Applets page.

Last updated by Mike May, S.J. on November 17, 2006.