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.
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.
Graphers
The 2D Grapher Applet
page collects 3 applets for graphing in the plane. One can either
use
the Cartesian Grapher to graph y as a function of x, or use the
Parameterized Grapher to graph x and y as functions of t, or use the
Polar Grapher to graph r as a function of theta.
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.
Continuity
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.
Differentiation
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.
Integration
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.
Sequences and Series
A theme of single variable calculus is how the Taylor polynomials
can be used to approximate functions. A Taylor
Polynomial explores the same themes.
Applets for
Multivariable Calculus
Theses applets are designed to be used for visualization
exercises in multivariable calculus.
Understanding
surfaces and graphs of several variables
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.
Vectors
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 R3, 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.
Continuity in several
variables
The Continuity
applet looks at
a
visualization of the definition of continuity as it applies to
functions of two variables. This is a generalization of an applet
that looks at single
variable
continuity.
Differentiation
of functions of several variables
The Partial
Derivatives applet finds partial derivatives as tangent lines to
the curves in slices where x or y is held constant.
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 R2 to R.
The Gradient-Contour
Applet uses the gradient field and contours to find local extrema
of a function of two variables. The Constrained
Extrema Applet addresses the problem when restricted to a
parameterized constraint function.
Integration in vector
fields
The Visual Divergence
and Curl Applet gives a representation of the divergence and curl
as you move around a vector field. The Divergence in 2-D Applet
compares the algebraic and integral definitions of the divergence of a
vector field in two dimensions. The Curl in 2-D Applet compares the algebraic and
integral definitions of the curl of a vector field in two dimensions.
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.