﻿ DEMOS with POSITIVE IMPACT NSF

DEMOS with POSITIVE IMPACT
NSF DUE 9952306

A project to connect Mathematics instructors with effective teaching tools.

David R. Hill, Temple University
Lila F. Roberts, Clayton State University

CALCULUS DEMOS

Click a on a topic to see a list of associated resources.

Functions, Continuity, and Derivatives

Derivative Applications

Integrals

Integral Applications

Partial Derivatives

Sequences and Series

Miscellaneous Topics

Functions, Continuity, and Derivatives

Generation of sine and cosine  Show the development of the sine and cosine functions and their graphs by 'wrapping' around a circle.

Domain and Range Graphically  This demo is designed to help students use graphical representations of functions to determine the domain and range.

for functions defined by formulas (animations)
for functions defined piecewise (animations)

Families of Functions and Curves  A toolbox of aids for teaching students about families of functions. The toolbox includes  collections of animations that illustrate how functions change when certain parameters are varied. The animations are designed to be run on a variety of platforms.

Constructing equations from Word Problems   This demo provides a toolbox of visual aids for geometrically oriented word problems.  These visual tools are designed to help students develop equations that provide an algebraic model for the problem.

Average Rate of Change  This demo provides students with a concrete understanding of the average rate of change for physical situations and for functions described in tabular or graphic form.

Escalator Motion and Average Rate of Change  This demo provides an early Introduction to related rates of change using escalator motion and average rates of change.

Jogger: Construct stories from time vs. distance graphs   The goal of this demo is to provide students with an opportunity to use information on average rates of change to create a story about the workout of a jogger from a time vs. distance graph of the jogger's run.

Continuity of piecewise functions  This demo provides a set of visualizations designed to help students better understand what it means for a piecewise function to be continuous at a particular domain value.

Definition of a derivative graphically  This demo can be used to investigate the approximation of the tangent line by a sequence of secant lines.

Piecewise functions: Investigating differentiability  This demo provides a set of visualizations designed to help students better understand what it means for a piecewise function to be differentiable at a particular domain value.

Sketch the Derivative from the Graph of the Function This demo provides instructors with interactive examples for the classroom or student assignments involving functions for which students are asked to sketch the derivative.

Trip stories: Create a story of auto trip from a time vs. velocity graph  This demo provides students with an opportunity to interpret graphical information from time vs. velocity graphs in order to create a story about an auto trip.

Sketch a Function from a Graph of its Derivative  This demo provides instructors with interactive examples for the classroom or student assignments for sketching a function given a sketch of its derivative.

Monotonicity: Algebraically and Graphically  This demo provides a set of interactive graphical visualizations designed to help students better understand what it means for a  function to be increasing/decreasing over an interval.

Cycloids  This demo uses parametric equations to graph cycloids.

Derivative Applications

Demos for Max-Min problems   This demo provides a toolbox of visual aids that illustrate fundamental concepts for understanding and developing equations that model optimization problems, commonly referred to as max-min problems. The focus is on geometrically based problems so that animations can provide a foundation for developing insight and equations to model the problem.

Demos for Related Rates problems  This demo provides a toolbox of visual aids that illustrate fundamental concepts for understanding and developing equations that model related rate problems. The emphasis is on the strategy to develop appropriate equations.

Cell Phones: signal-to-interference ratio  This demo develops a function that measures the power of the signal of a cell phone as a user moves in a cellular network and then determines the position in the network when the signal is a maximum.

Box MaxMin Problems  This demo provides practice with optimization problems related to geometric figures. A sequence of "box problems" is designed to help students develop an equation for the volume of a box constructed from a rectangular sheet of cardboard by cutting away portions and folding the remaining portions to construct a box in a particular way.

"Inverse" Box Problems  This demo provides practice with word problems related to geometric figures. The sequence of "box problems" was designed to give students practice interpreting the given geometric characteristics of a box in order to determine the smallest rectangular piece of cardboard from which the box could be made.

Integrals

Area between curves as a limit of Riemann sums  This demo combines the visualization of approximating rectangles together with a graph of the approximate areas as a function of the number of rectangles.  Thus, the limiting behavior of the approximating sums can be observed on the graph.

Riemann sums: a symbolic and graphical approach  This demo uses a computer algebra system to investigate various approximations to the definite integral. Right hand endpoint, left hand endpoint and midpoint Riemann sums as well as trapezoidal and Simpson's rules are expressed in closed form as functions of
. Behavior as approaches 0 is observed.

Integral Applications

Visualizations for Volumes of Solids in Calculus  The purpose of this demo collection is to help students to understand the concepts that motivate the elements of computation of volumes of solids.

Volumes by Section  This demo deals with solids that can be sliced into pieces with a known cross section.

This demo focuses on solids generated when a planar region is revolved about the x-axis.

Solids of Revolution: The Method of Shells This demo involves revolving a region bounded by a curve y = f(x) and the domain interval [a,b] about the y-axis.

The Washer Mehod for Volumes of Solids of Revolution This demo involves revolving a region about one of the coordinate axes. In this demo the resulting solid has a "hole".

My Favorite Mug  A physical demonstration that involves the approximation of an integral with hands-on measurements.

Interactive Area and Arclength Approximations Using Flash  The goal of this demo is to provide instructors and students with interactive tools for approximating area under a curve and arc length using elementary numerical methods.

Volumes of physical objects  The purpose of this demo is to illustrate the use of the method of cross sections to find estimates of the volume of physical objects.  The objects in this demo are not solids of revolution.

The utility of catenaries to electric utilities    The purpose of this demo is to illustrate how hyperbolic functions and arc length integrals are used to model hanging cables.

Enlightening Volumes: curve fitting to approximate volumes  The purpose of this demo is to illustrate to students that techniques used to compute the volume of solids of revolution can be applied to real objects.  The demo employs digital photography and various curve-fitting techniques to approximate functions that, when revolved about an axis, yield a solid that approximates the object.

Partial Derivatives

Partial Derivatives, geometrically  This demo provides a visual foundation for partial derivatives of functions of two variables, z = f(x,y).

Sequences and Series

Mullikin's Nautilus  To demonstrate limits geometrically, first on the real line and then in the plane.  We begin with an example on the real line that motivates the extension to the plane.

A "sweet" introduction to infinite series  This demos introduces the main ideas and vocabulary of infinite series and the convergence of series using "food".

Miscellaneous Topics

Constructing conic sections on a white board   This demo provides a visual development for the "locus of points" definitions of the conic sections.

Logistic Curve  This is an interactive demo that illustrates the generation of a logistic curve. It is quite effective  immediately after  a discussion of inflection points.

Centroids (without calculus)  This demo demonstrates a simple physical method of determining the centroid of an irregular region.

This demos provides a visual foundation and geometric intuition for least squares models of data sets of ordered pairs using lines or parabolas.

Taylor polynomials - a visual approach to approximations  The purpose of this demo is to use a graph of the function y = f(x) and its nth Taylor Polynomial, pn(x) to illustrate the approximation of y = f(x) by a Taylor Polynomial centered at x = a. We include an option for the visualization of the error function  Rn(x) = f(x) - pn(x).

Inviscid flows: Go with the flows  This demo provides a collection of examples of physically significant vector fields that can be used to illustrate important topics in the study of vector calculus.

DRH 9/10/2010          last updated 9/15/2010 DRH

Visitors since 9/15/2010