# Math Applets for Calculus at SLU

These following collection of applets are designed for use in calculus courses.  Another page collects applets for courses below calculus.  Some of the applets were developed at SLU and some have been developed elsewhere and are included by permission.  If you would like to host the applets locally, please contact Mike May, S.J.

### Applets for single variable calculus

Preliminary material
Graphers
Continuity
Differentiation
Integration
Sequences and Series
General

### 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

When working through the understanding of various kinds of functions it is useful to be able to graph a function with parameters a, b, and c, in the definition of the function, with the parameters controlled by sliders.  Moving the sliders lets the student explore families of functions.  The Families of Functions Applet is a GeoGebra applet for looking at the graph of such a family.  Similarly, the Family of Graphs Applet is a JCM applet designed to look at families of functions.

The Function Composition Applet is a JCM applet that links together the graph of two functions with the graph of the function defined by the composition of these two functions.  The Visual Composition of Functions Applet is a Banchoff applet that 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.

It is often useful in calculus to review the basic ideas from trigonometry. The Trig Functions Applet is a Geometer's Sketchpad applet that connects the 6 standard trig functions with lengths of line segments from a  diagram connected to the unit circle.  The same idea is covered with a Trig Review Applet using GeoGebra.

### 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 Epsilon Delta Applet is is a JCM applet designed for a visual exploration of the delta-epsilon definition of continuity. The user highlights an epsilon and delta band around a proposed limit of a function at a point.  It is easy to zoom in or out, and the applet has a nice collection of pre-set examples.  A second applet also looks at the delta-epsilon definition of continuity of functions in one variable.  It is not as robust as the JCM applet, but it generalizes to a applet looking at the delta epsilon definition of continuity for functions in two variables.

### Differentiation

A standard demonstration in calculus in showing that the tangent is the limit of a series of secant lines.  This is implemented in a number of applets, each of which has it particular strengths.  The Secant Tangent applet was developed locally.  This applet also computes a numeric derivative and allows the users to plot their guess of the derivative for comparison.  The GeoGebra Secant to Tangent Applet lets one easily control the base point and the value of delx in the secant by sliders.  The JCM Secant Tangent Applet  has a number of nice pre-loaded examples and nice graphics.  We also have a a Banchoff  applet that looks at secant lines converging to a tangent line.

The Slopes and Derivatives applet is a GeoGebra applet that traces out the slope of the line tangent to a curve as the base point is moved along the curve.  One can compare this to a guessed derivative.  The First Derivatives Applet is a JCM applet that ties together the graph of a function with the graph of its first derivative.  (The function and its derivative are plotted in side by side windows.)  A slider moves synchronized points on the two graphs.   A JCM Second Derivative Applet ties together the graph of the function with the graphs of the first and second derivatives.  (The function and its first and second derivatives are plotted in side by side windows.)  A slider moves synchronized points on the three graphs.

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

The Newton's Method page illustrates the use of tangent lines to find roots of functions.  It also gives examples where the technique fails.  This version focuses on moving through the method in a step by step fashion.  It lets the student focus on how the method works. A second applet using the Newton-Raphson method lets you compute up to 200 iterations.  This applet lets the student focus on questions of when the method fails.

The JCM Chain Rule applet is a modification of the Function Composition Applet. It show that the derivative of the composition of functions is the product of the derivatives taken at the appropriate points.  Similarly, the Visual Chain Rule Applet is a modification of the Banchoff applet for visual understanding of differentiation of a composition of functions.

A single variable differentiability applet explores differentiability as an extension of continuity.  Rather than working with a box, you work with a cone that surrounds the proposed tangent line.  This applet extends to an applet looking at differentiability of functions of two variables.  This is a Banchoff applet.

### Integration

The Riemann Sums applet is designed for a visual exploration of Riemann sums and the relation of these sums to anti-derivatives.  It computes 6 different types of "Riemann sums", plots a numerical antiderivative, and allows users to plot their guess at the antiderivative for comparison.  The JCM Riemann Sums Applet also explores how Riemann sums converge to the definite integral.  It does not have all the features of the local applet, but the graphics are smoother and faster.

One of the standard applications of the definite integral is to find the Area Between Two Curves.  This Banchoff 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 GeoGebra Taylor polynomial applet lets you compare a function with its Taylor polynomial approximation and the error term of the difference.  The base point and degree are controlled by sliders.  A locally developed Taylor Polynomial Applet is designed to explores the same themes.  These applets also give formulas for the approximating polynomials in terms of  x-x0.  A Banchoff Taylor Polynomial explores the same themes.

The Sequences and Series Applet allows for visual exploration of sequences and series.  It plots a geometric, harmonic and exponential series as well as two user defined series.  The user can choose to plot terms, sums, or ratios.  The applet plots 100 points at intervals of up to 1000 steps.  It gives values at specified steps.

The Series Functions Grapher Applet graphs functions of x that are defined as a summation over k of terms in k and x. It is useful for exploring Taylor and Fourier series.  The applet allows the user to plot a function defined by a series, as well as a closed form function for comparison, along with the difference of the two.

The JCM Series Grapher Applet graphs functions defined as series over k of terms in x and k.  It is similar in functionality to the Series Function Applet above.

### General

Another nice Applet done by David Ecks is the xFunctions applet.  It has seven screens, each of which is a nice applet.

## Applets for Multivariable Calculus

Theses applets  are designed to be used for visualization exercises in multivariable calculus.  They are local implementations of an applet written at Brown University for the calculus class of Tom Banchoff.  The applet is used with permission.  The applet seems to work best with Internet Explorer.

### 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 Applet is a Geometer's Sketchpad® Applet. It lets you visualize two vectors with their sum and difference and the vectors measured both in polar and rectangular coordinates. The Adding Vectors in 3D Applet is a Banchoff Applet that lets you move the visualization to 3 dimensions.

The Projections, Dot and Cross Products Applet is a Geometer's Sketchpad® Applet. It lets you specify two vectors by dragging their endpoints.  It then gives bit a visualization as well as numerical data for the projection of one vector on another, their dot product of the two vectors, and the cross product of the two vectors. with their sum and difference and the vectors measured both in polar and rectangular coordinates.

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 Fields Applet plots a vector field defined by a pair of functions in x and y.  This is another JCM applet.
We also have a Banchoff Applet that plots Vector Fields in either 2 or 3 variables.

The Integral Curves Applet modifies the JCM Vector Fields Applet and allows the user to plot integral curves to a vector field.

### 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 applet is a Banchoff Applet that 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 is a  Banchoff Applet that 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.

Several of the applets above are referred to as JCM applets.  This refers to an NSF sponsored  project,  the Java Components for Mathematics project at Hobart and William Smith Colleges., that produced a series of JCM-Applets that are quite useful for calculus and pre-calculus.

Several of the applets above are referred to as Banchoff applets.  This refers to work done at Brown University under the direction of Tom Banchoff.  The project produced a demonstration applet that is quite flexible and can be configured for demonstrations and projects.  The configurations can be saved as a set of parameters used in calling the applet.  These applets are collected on the Banchoff Applet Page.

Several of the applets above are referred to as Geometer's Sketchpad® Applets.  That means that the applets were created with Java Sketchpad®, a component of Geometer's Sketchpad®.  More accurately, they were created as a set of parameter's that can be used to call the Sketchpad® Applet in a particular configuration.  These applets are collected on the Geometer's Sketchpad® Applets page.

Several of the applets above are referred to as GeoGebra Applets.  That means that the applets were created with GeoGebra, an open source program created by Markus Hohenwarter.  The program is available from http://www.geogebra.at/.  They applets are more precisely configurations of a master applet.  These applets are collected on the GeoGebra Applets page.

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Last modified May 14, 2008 by Mike May, S.J.