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
a JCM applet designed to look at families of
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.
The Epsilon Delta
is a JCM
applet designed for a visual exploration
of the delta-epsilon
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
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
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
a number of nice pre-loaded examples and nice graphics. We also
have a a Banchoff
applet that looks at secant
converging to a tangent line.
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
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
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.
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.
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 two variables. This is a Banchoff
The Riemann Sums
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
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
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
A second application is to find the length of a curve with the curve
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 GeoGebra
lets you compare a function with its Taylor polynomial approximation
the error term of the difference. The base point and degree are
controlled by sliders. A locally developed Taylor
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
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.
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
graphs functions defined as series over k of terms in x and k. It
is similar in functionality to the Series Function Applet above.
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
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
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.
Approximation applet is a Banchoff
that explores the issue of the region where the tangent plane can be
used as a good approximation of a function.
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.
Return to the courseware page at Saint Louis University
Last modified May 14, 2008 by Mike May, S.J.