Java applets for
First Year Calculus
If you have a computer and browser which support Java, there are
several useful programs you can run from the Web to help with
calculus. These do not require you to know any programming
or anything about the Java language.
Applets Made Specifically for use with Georgia Tech Courses
A collection of applets has been written specifically for use in the
courses in the Georgia Tech calculus sequence. This growing collection is cataloged here by
course, together with a collection of suggested projects and experiments for
students to do. Each experiment has specific learning objectives that are described
in the project catalog.
Applets from outside Georgia Tech
Plotting graphs
(an applet by
Scott Herod
at Boulder.) A picture is worth a thousand data points. With this applet
you can plot simple functions on line.
Tangent lines and secant lines
(an applet by
Bill Ziemer at
Long Beach.) This applet gives a visual idea of the tangent line as a
limit of a secant line.
Maple explorations
for
Differential Calculus
The following items are a collection of mathematical explorations
developed
for Georgia Tech's Math 1507. They are all formatted with
Maple V. Release 4.
These worksheets are ready to use, without much prior knowledge of Maple.
As you read through them, you simply have to press ENTER to perform the
calculations, and if you don't know Maple already you will easily
begin learning how it works. After reading through a given worksheet,
you can generate your own models and alternative scenarios by
cutting and pasting Maple code.
In each worksheet a
model is introduced with a description of the content and of the
level of presentation.
These may be down-loaded onto your computer as text files,
opened with Maple, and saved as active worksheets. Alternately,
you may configure your Web browser to launch Maple as you choose a
file.
An Introduction to Maple:
A modest
worksheet to get started, by showing how to define
and plot functions.
Drawing Graphs:
One of the primary uses of the computer will be to give visualization. This worksheet gives syntax for drawing graphs. The worksheet does not use the calculus. Rather, it is an introduction to techniques for drawing graphs with Maple.
Interpreting Graphs:
The goal of
this
worksheet is to understand the graph of the following function:
> h:=x->(x^3+3*x-5)/(x^2+2);
Although you can just plot it in Maple, we want to do this
systematically to get a better idea what to look for when looking
at the graph. Read each question, and try to see if you could have
guessed this by looking at the graph of the function.
Parametric Animations: Animation is provided for graphs constructed from a + cos(theta) as the parameter a changes in time. The worksheet is appropriate at a precalculus level. It is useful for studies in animating graphs in polar coordinates.
Fitting Data with a Periodic Character: Often, data that arises from various geophysical properties has a periodic character. For example, take the length of time from sun-up until sun-down. This time difference does not vary appreciably through the years. Thus, one might fit the data for computations of this periodically varying observation with trigonometric functions. This worksheet fits such data with least square fits, but using trigonometric functions.
Gas Mileage Analysis: Gas mileage data is provided from a 4000 mile trip. The worksheet computes linear and quadratic regression fits for the data. Interpretations for the surprising results are requested. The worksheet is appropriate at a precalculus level.
Slopes, Tangent Lines, and Derivatives: This worksheet gives animated visualization for the computation and role of a tangent line. The worksheet is appropriate at an early stage in a study of differential calculus.
Reflections on Differentiation:
Here
Maple's ability to differentiate both functions and expressions
is introduced. The problem of finding extreme values of simple functions
is illustrated by using Fermat's Principle to find the path of reflected
light. Some fancier plotting capabilities and also the equation
solving ability of Maple
are also introduced.
Human Height/Weight Relationships: Height/Weight data for humans suggest how one might expect the weight of humans to increase with increasing height. A little reflection on how the volume of a sphere or cube increases with height suggest this relationship for humans should not be linear. But, what should it be? This question is explored in the worksheet.
Maximum Heart-Rate for a Runner: The heart's pulse-rate for a well-trained runner will increase at the beginning of a run, and then decrease as the runner experiences a "second wind." There after, the pulse-rate will gradually increase as the runner switches from metabolizing glycogen to converting fat into an energy source. This toy model mimics this phenomena. It asks when the maximum heart rate will occur. The answer is obtained with the standard tool: set the derivative equal to zero and solve. It may be that the "solving" is the most interesting part from the perspective of computing in this worksheet.
A Model for Cooling: A hot pitcher of water was allowed to cool in an environment which was below freezing. We provide the outside temperature, the temperature of the water every three minutes over an hour's period, and assume Newton's Law for cooling to make an analytic fit for this data. This model uses the one-dimensional calculus to get the fit for this data. The fit is designed to minimize a standard measure of error between the data and the analytic model.
Solving Equations:
This
worksheet
is about continuity--the intermediate value property of continuos
functions, etc. Equations are solved using "bisection,"
regula falsi, and Newton's method. This assignment also introduces the
student to Maple procedures.
Solving f(a) = b: The Method of Bisection: The Method of Bisection for finding roots of an equation is not conceptually hard. It has two especially good features: first, it is easily understood, and second, it can provide initial guesses for methods which might converge faster -- such as Newton's Method. We illustrate the method and provide an example to show that a little care must be taken.
Solving f(a) = b: Newton's Method: Newton's method is a standard application for the first quarter differential calculus. In this worksheet, we illustrate the method for finding roots of an equation and then provide an example which has enough pathology to be interesting.
The Year of the Fastest Growth for the US Population
: The U. S. Constitution requires a census every ten years.
We provide a "Logistic Fit" for that data and graph the fit superimposed
with the data. This
worksheet
then asks the following question: In what year does this logistic fit
suggest the fastest growth in the U. S. census data?
Scores, Grades, and Deviations:
Students always want to know
how their grades are determined and how well other students are doing in their class. This
worksheet provides a
way to think of this question. Histograms and normal distributions are drawn. Data from
the class will be provided.
Hyperbolae and Navigation
:
The Loran-C navigation system is used to illustrate the application
of properties of the hyperbola to positioning of ships, aircraft, etc.
in this
worksheet
.
There is the briefest of introductions to using Maple to solve
a system of equations.
A roller coaster
:
A mythical roller coaster is one kilometer long, and takes three minutes.
In
this worksheet your
location is given as a function of time and you are asked various
questions about the motion which can be answered with Maple.
Riemann sums
(an external link to an applet by
Bill Ziemer at
Long Beach.) This applet allows the user to choose various functions and
values of dx.
Moments
and centroids
(an external link to Project Links) This applet illustrates the
principles of mechanical balance.
Graphical Solutions for Two Dimensional Differential Equations
(an external link to an applet by
Scott Herod
at Boulder.)
Quick, graphical solutions for a two dimensional system of differential
equations are only a point and click away.
Chemical kinetics
(an external link to Project Links). This applet is concerned with
differential equations describing rates of chemical reactions.
The following items were also developed for
Georgia Tech's Math 1508. They are
similar to the
materials for differential calculus
formatted with Maple V. Release 4.
Area by Chance: The Riemann integral is usually defined as the limit of a collection of approximating sums. There after, the fundamental theorem of integral calculus provides methods for evaluating integrals without computing limits of sums. As an alternative idea, this worksheet introduces a random number generator and what is usually called "Monte-Carlo Techniques" to evaluate integrals. "Area by Chance" is a good worksheet to examine early in the introduction of the integral.
Important geometric relationships exist between the graphs of f, the derivative of f and the integral of f. This worksheet compares their graphs and asks the user to identify which is which in an overlay of the three graphs.
A Few Moments with the Cosine Function
:In this worksheet, it is observed that the graph of the cosine function on the interval [-Pi/2, Pi/2] looks similar to the graph of an inverted quadratic polynomial. We present five methods for finding quadratic approximations for the cosine function on this interval. Techniques use simple differential and integral calculus. The worksheet ends with a polynomial of degree 6 that approximates the cosine function on the interval [0, 4 Pi] with surprising accuracy.
Techniques of Integration: Substitution and Integration by Parts: It used to be that a calculus class would study the techniques of integration so well that the student could work out how to integrate functions such as x arcsin(x). The computer can be used to help with the calculus when methods such as substitution or integration-by-parts are correct choices for integration techniques.
Rotations
: When drawing a figure of revolution, there are several ideas that arise: What will the figure look like? What is the area of the resulting surface? What is the volume of the enclosed solid? All three questions are addressed in this worksheet.
A Distribution of Weights
:
Most of us have experiences with balancing a series of weights on a line segment. The balance point is located at a point which may be computed as a combination of sums and products of weights and distances from the balance point. For the purposes of the sciences, we often consider a solid as being concentrated at a single point. The location of the center of mass of a continuously distributed weight is computed in this worksheet.
Length and speed along a curve
:
How can we calculate the length of a curve, whether given as a graph y = f(x)
or as a parametric curve? If an object moves in a specified way along
a path, how fast is it moving? This
worksheet
shows how to calculate the answers and sets Maple up to help do so.
There are also some supplementary exercises. (Also available as
plain text.)
You may wish to recall the worksheet on the roller
coaster at this stage.
A Fair Shake
:
Suppose you throw a pair of dice 360 times. How many times the sum of the top faces will be five can be predicted, or any other sum for that matter. It would be curious to actually do this. That's what we do in this worksheet -- only Maple does the throwing and the counting. A histogram is made for the results so that the result can be compared with a normal distribution. Also, the extent to which the normal distribution predicts the result is found by integrating this distribution.
Hot Wheels
:
Suppose a ramp is constructed from a point A to a lower point B, that the ramp stays in a vertical plane containing A and B, and that a cart is allowed to run down the ramp (without friction). What is the shape of the ramp to allow the cart to arrive at the bottom in the least time? This classical problem is introduced in this worksheet.
The Hausdorff Moment Problem
:
We approximate the cosine function on an interval with a polynomial that has the same first seven moments as the cosine function does. The ideas arise from the classical moment problem.
The Theorem of Pappus
:
Suppose that the plane region R is revolved about the line L in the xy-plane. Suppose also that the line does not intersect the region. The volume of the solid generated is equal to the product of the area of R and the length of the circumference of the circle traced by the centroid of R. This result of Pappus is illustrated in this worksheet.
Polynomial Approximations
:
If f is a function with N derivatives at the point c, then it is not so
hard to give a polynomial which has the same N derivatives at c that f has.
We ask: Suppose we select an interval on which f is defined. How do we
choose c so that the polynomial described above approximates f on the
specified interval best? This
worksheet
will suggest an answer for this question with a specific example.
Growth, Decay, and Exponential Functions
:
The real reason we learn about the exponential function in this course
is that we need it to solve important differential equations, like
those which describe growth or decay. In this
worksheet we discuss some differential equations and their applications to
archaeology (carbon dating), atmospheric radiation, and predictions of
populations. Related worksheets include the one on
Newton's law of cooling and the one on
U.S. population data.
Explorations for
Multidimensional Differential Calculus
The following items are a collection of mathematical explorations
developed
for Georgia Tech's Math 1509.
Three Dimensional Graphics.
Maple can be used to help visualize the shapes of surfaces, as well
as for the
computation of algorithms. Some tools are suggested in these two
worksheets. The
first one illustrates the commands with straightforward
examples, while the
second one develops some graphics tools in a specific context.
Projections Onto Lines and Planes.
Given a line (or a plane) and a point A not on the line (or plane),
how do you find the point on the line (or plane) that is closest to A?
The dot product provides a good computational way to do this.
The techniques of this
worksheet
generalize to more complicated situations in many dimensions.
In fact, the methods are independent of the dimension.
The Volume of a Parallelepiped.
The area of a parallelogram and the volume of a
parallelepiped can be found using dot-products and cross-products. The
worksheet
makes clear the ease of finding these geometric quantities without
resorting to angle measure. It's all in the arithmetic!
Tangents, Normals, and Curvature.
Perhaps as much as any other place, the computations for the
tangent and normal to a space curve are best accomplished
with a computer. The calculus to obtain these geometric notions
for even simple functions can be formidable. Because these ideas
arise so prominently in the task of resolving both motion and the
forces causing motion, it is well that the ideas be understood at the
beginnings of a study of multidimensional calculus. We will give
computational tools and illustrate the ideas with examples.
A Water Whirl.
If a cylinder is partially filled with water and whirled about the major axis, the surface of the water changes shapes depending on the speed of the revolution. This worksheet
resolves the forces acting on the water into components and determines the
shape of the spinning surface.
Parabolic and Spherical Mirrors.
A
classroom project
on this subject by
Xu-Yan Chen. No software is
required to read this link, although you may wish to look at the
Maple worksheet
on the
water whirl, which is an active document with related
calculations.
Tangent Planes and Normal Lines.
This
worksheet
will provide an explanation for how a tangent plane and a normal
line for a surface S can be constructed in case f
has continuous partial derivatives in x and y. Graphical illustrations are
provided. Also, a pathological example is given to
illustrate when the procedure can fail.
Surface Area and Pediatric Pharmacology.
Medical technologists often decide dosage levels for medication based on
the size of the intended recipient. In particular, if the drug is to be
administered intravenously, the surface area of the individual is
important. This
worksheet, provides a correlation between height, weight, and surface
area for humans as determined by the commonly used West Nomogram.
Additionally, partial derivatives (and the chain-rule) are used to
find the rate of change of the surface area for a rapidly growing adolescent.
Newton's Method and Newton's Law of Cooling.
Real data are given for a warming body. A
model
is provided for the rate of warming. The task of finding the
coefficients for the
model requires the solution of a nonlinear equation.
For this latter, we use the
nonlinear, multidimensional Newton's method for
finding a zero of a function.
Maximization, with Constraints.
A common tool for finding the maximum of a function with constraints
is to use the method of Lagrange multipliers. In this
worksheet
we not only give algebraic tools for solving the equations associated
with the method, but also illustrate the geometry that serves as the
motivation for the method.
Cobweb diagrams
It often happens in science, mathematics, and engineering that is on
interest to find a fixed point of a function. In this
worksheet
we investigate how the process of seeking such fixed points through
iteration can be visualized geometrically. The iteration
process is called the cobweb construction.
Explorations for
Multidimensional Integral Calculus
This subject is studied in the second year of calculus, and
online materials are available on the Web page,
"The Second Year of Calculus."
Some external links
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