# Banchoff Applets

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.

### Applets for single variable calculus

Preliminary material
Graphers
Continuity
Differentiation
Integration
Sequences and Series

### Applets for Multivariable Calculus

Understanding surfaces and graphs of several variables
Vectors
Continuity in several variables
Differentiation of functions of several variables
Integration in vector fields

## Applets for single variable calculus

### Preliminary material

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.

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