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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 arclength using elementary numerical methods. In put consists of a
domain and function formula. Then the user selects points that need not be
equispaced along a curve. (The points can be in any order.) The software then
plots line segments for arclength approximation, forms trapezoids for the
trapezoidal rule, or displays parabolic arcs for Simpson's rule and then
provides a numeric approximation. These routines provide an opportunity to form
estimates for functions which when used in integral expression have no closed
form antiderivative. They also provide an opportunity to relate the geometry of
the curve to adaptive numerical approximation. (This is a beta version of
this demo. Please send comments. 7/14/2008.)
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DEMOS for Related Rates
In the gallery of related rate animations we have included three new demos:
<> Filling a conical tank with a fluid. We simultaneously fill the inverted cone
and show a corresponding triangle that simulates the change in height and radius
of the cone of fluid.
<> Searchlight rotates to follow a walker. We show a figure walking at a given
rate and the beam of light following the figure. A triangle with changing
hypotenuse is displayed along with a table of information.
<> Filling a cylindrical tank, two ways. We compare filling a standard cylinder
and the same cylinder laying on its side. As the fluid level rises we display a
table of information about the volume, height of the fluid in each cylinder and
the rate of change of the height in each cylinder. Graphs are also generated.
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Matrix Algebra Demos
Nine additional demos have been include in this collection, including
inverse, linear combination, matrix transformations, span, closure and linear
independence/dependence. Some of the demos now include a quiz or exercises to be
done off-line. (8/25/2007)
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DEMOS for
MAX-MIN Problems
In the gallery of many of the demos now have interactive Excel files for
estimating the optimum value. These files can be downloaded for classroom use by
instructor or for student experiments. (6/6/2007)
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"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. We present a sequence of geometric problems in which the
object to be modeled is a
box to be constructed in a particular way from a rectangular piece of
material. Since the
dimensions of the box are specified we are not solving an optimization
problem involving volume, rather we
want to develop an algebraic model that can be used to determine the
dimensions of smallest rectangular piece of cardboard that can be
used to construct the particular box. Only algebra is required. Students must interpret geometric information given for the box in order to
appropriately assign values to portions of the rectangle that circumscribes
the unfolded box. (5/30/2007)
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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. The equation for the
volume is to be formulated in terms of the dimensions L and W of the piece of
cardboard and a parameter x, often related to the height of box, so that
calculus can be used to determine the value of the parameter that will maximize
the volume. An Excel file accompanies each type of box and provides a geometric
model. The dimensions of the piece of cardboard can be changed and the volume of
the largest box approximated. (5/30/2007) (This is a 'beta' version; please send us comments.)
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Trip
Stories
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. Students should be familiar with the
concept of the slope of a line segment so that the velocity of an object can
interpreted.
Included is an Excel worksheet with six auto trips represented by time vs.
velocity graphs. The Excel worksheet is compatible with PC and MAC. In addition an
animation in both Flash and QuickTime formats.
(1/28/2007)
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Jogger: Time
vs. Distance Graphs
This demo provides 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. Students should be familiar with the concept of slope of a line, computing the slope of a line,
and average rate of change.
Included is an Excel worksheet with six jogger paths. The Excel worksheet is
compatible with PC and MAC. In addition an
animation with audio in both Flash and QuickTime formats.
(1/11/2007)
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Sketch the Function from a Sketch of its Derivative
This demo provides instructors with
interactive examples for the classroom or student assignments that ask for a
sketch of a function given a sketch of its derivative. Techniques for
determining the type of sketch to generate, animations (using Flash and QuickTime), and 10 Excel routines which
the derivative of a curve and simultaneously
generate 3 possible sketches for the function are included. (1/2/2007)
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Sketch the Derivative
This demo provides instructors with
interactive examples for the classroom or student assignments involving
functions for which students are asked to sketch the derivative. Techniques for
determining the type of sketch to generate, animations (using Flash and
QuickTime), and 10 Excel routines which sketch a curve and simultaneously
generate 3 possible sketches for the derivative are included. (12/29/2006)
