# Studies in the First Two Years of Undergraduate Mathematics

## Mathematical explorations formatted with Maple 6

Each model is introduced with a description of the content and of the level of presentation. The worksheets are formatted as Maple 6 documents. 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. After opening the files with either method, they may be modified to generate different models with different parameters.

## Differential Calculus

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.

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.

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 theworksheet.

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 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?

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.

## Integral Calculus

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 alternate 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.

Graphs of f, f prime and integral f.

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.

Techniques of Integration: Substitution and 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 rovolution, 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.

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 Hausdorf Moment Problem

We approximate the cosine funtion 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.

## Multidimension Differential Calculus

Three Dimensional Graphics

Maple can be used to give good intuition for the shape of surfaces, for the calculus on multidimensional functions, and for the computation of algorithms. Some tools are suggested in this worksheetR5 or worksheetR4 . The tools are put in a context, but it is the introduction to using Maple to draw surfaces that is of primary importance in this assignment.

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 thisworksheetR5 or worksheetR4 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. TheworksheetR5 or worksheetR4 makes clear the ease for 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 normalR5 or tangent and normal R4 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 worksheetR5 or worksheet resolves the forces acting on the water into components and determines the shape of the spinning surface.

Tangent Planes and Normal Lines

This worksheetR5, or 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 worksheetR5 or 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 Used with Newton's Law of Cooling

Real data is 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 non-linear, multidimensional Newton's method for finding zero's of a function.

Cobweb Mappings

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 a Cobweb Construction.

Maximization, with Constraints

A common tool for finding the maximum of a function with constraints is to use the method of La grange 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.

## Multidimensional Integral Calculus

Multidimensional Data

Most often, it is desired to apply techniques of the calculus to observations which have been made as data. This worksheet changes multidimensional data into multidimensional functions. Calculus techniques can be applied to these functions. The illustration is taken from data relating the circumference of the thigh, the ability to leg-press weight, and the ability to leap. We ask, how high might you expect the average person with a thigh circumference of 53 centimeters to be able to jump if he can press between 100 and 150 pounds?

Intersecting Cylinders

The problem of finding the volume of two intersecting cylinders is one of those classical problems for three dimensional integration that most students who pass here encounter. We suppose that the cylinders have radius 1. The intersections of the cylinders with the x-y plane and the y-z plane is a circle with center the orgin and radius 1. This worksheet gives visualization to the intersection of the cylinders

Centroids for Morphing Regions in the Plane

This worksheet is concerened with centroids for curves and regions in two dimensions. Questions are asked about how centroids move as the regions grow or shrink.

A Potential for Being Conservative

This worksheet presents the criteria for a function F to be a conservative field, as well as implications of this condition. One implication is to the evaluation of line integrals, the other implication provides a method to check for F being conservative.

The Area of Familiar Surfaces

The subject of this worksheet is surface area. Techniques for computing the surface area of graphs progress from those for planes, to graphs of real-valued functions defined on the plane, to surfaces defined parametrically. The student is asked to get the surface area of a torus.

Flux in Three Dimensions

In this worksheet we take one field and compute the flux across three surfaces, all having a common boundary. The reader is asked to compute the flux across a fourth surface which has the same boundary.

An Application for the Divergence Theorem.

In this worksheet we compute the flux through a cylinder immersed in a field. The calculations are done twice to provide verification of the Divergence Theorem for this application.

Three examples, two parameterizations.

Sometimes students have trouble with switching to polar coordinates. Some classes wonder "What's the issue?" In others, some students easily talk others into being confused. If you are a little confused, this worksheet may help you. If you've got it all straight, this one may be a bore.

## Linear Algebra

The Downtown Rental Car Firm

This worksheet uses linear algebra to solve a delivery problem. The concepts of eigenvalues and eigenvectors are used. Regular transition matrices are introduced.

## Mathematical explorations appropriate for

Maple and Differential Equations

Students who have used Maple through the calculus will have a good intuition for how the Maple syntax works. This worksheet provides some elementary examples for what can be done in differential equations with Maple. The elementary methods are designed to give understanding for analytic and numerical solutions of differential equations.

Eat and Be Eaten

On page 86 of the February 7, 1998, issue of Science News (Vol. 153) there is an article titled "How low will we go fishing for dinner?" This article explains that life in the oceans might be divided into five trophic levels, each a predator on the one below. Typically, humans have fished from trophic level three. More recently, the authors of this article argue, as we deplete level three stock, we are willing to eat from further down this division of fauna. This worksheet is a simple multi-species predator prey model suggested by the Science News article.

Spruce Budworm Predation

This example explores a model for the growth of an infestation of a spruce budworm population in a Balsam Fir forest. We suppose the growth of the budworm to be of logistic type, except we add predation by birds. In this example, it becomes clear that the long range forecast for the model is strongly influenced by small changes in the parameters.

Coffee With the President

The President and the Prime Minister have coffee together. One of them pours in cream immediately, the other waits ten minutes. Which has the cooler cup of coffee? This problem is classical in undergraduate differential equations. A slight twist is added to the problem because of a bright Georgia Tech graduate serving as a Presidential Aide.

A Logistic Fit for the U. S. Population data

A common method used to model the increase in size of a population whose growth is limited by population density factors is to create a logistic model. We illustrate here how to make a logistic model for the U. S. Population data. This model is contrasted with an exponential fit for the data.

Flow Across Membranes

This worksheet models the flow of a fluid through a cylinder whose sides are a thin walled membrane. The membrane permits the absorption or release of a solute. Whether the solute is absorbed or released depends on the difference in the concentration in the cylinder and in the surrounding medium. With the assumption that the cylinder enters regions that alternately have higher or lower concentrations than the fluid in the cylinder, we seek a description for the concentration in the cylinder.

Flow Across a Membrane with Con-current or Counter-current Flow

Imagine two tubes, one with a smaller radius and contained in the other. Also, imagine the walls of the smaller tube being a membrane permeable to a solute that is dissolved in a fluid flowing through the smaller tube. A "cleaning fluid" flows in the larger tube. This worksheet explores the difference in the efficiency in removing the solute from the inner fluid depending on whether the fluids flow in the same or opposite directions.

Mixture Problems

A contaminant flows from one body of water through a series of other containers. We ask how long it will take the input in the first container to have a significant effect on the containers which follow. There is a time delay for when the mixture peaks in containers down the line. This problem introduces ideas surrounding compartment models.

Pharmacokinetics

We examine a simple illustration of a model for the passage of a pharmaceutical into the circulatory system through the gastro-intestinal tract. The model assumes a drug is taken every six hours. The reader is asked to buffer the drug, delaying its absorption into the circulatory system, and to illustrate graphically what the changes will be.

A Model for the Movement of a Disease Within a Population

This model is commonly used in introductory discussions of the movement of a disease within a community. There are two suppositions. The first is that an infectious disease has infected a group of people in a community. The second supposition is that the people in the community can be divided into the susceptible class, the infected class, and a recovered class. The dynamics begins when the small group of individuals who are infected with the disease come into contact with the larger population. This worksheet includes the possibility that the recovered class moves back into the susceptible class after some fixed time period.

Phase Portraits for Linear Systems in the Plane

Phase portraits for systems of differential equations give information about the character of solutions for the system. Such information is especially important in case analytic solutions can not be found. The notions for drawing phase portraits are introduced for simple, linear systems. An animation is includedin this worksheet .

Phase Portraits for Nonlinear Systems in the Plane

In this worksheet, we examine the local behavior of a nonlinear system near a critical point. This process will involve the following steps: Locate the critical points. Classify the configuration and stability of the critical points. Make whatever statements that can be made about solutions for the differential equation that start near the critical points.

Getting Control with Linear Algebra

This material explains the linear algebra tools necessary to solve a simple control problem. The set up is as follows: specify a second order, constant coefficient differential equation with initial conditions. Also, specify where you want the solution to be at a particular time. A forcing function is found which drives the solution to the place you specified at the time you specified. A Maple worksheet is provided to carry out all the computations.

Graphical Solutions for Two Dimensional Differential Equations

Quick, graphical solutions for a two dimensional system of differential equations is only a point and click away. The program is fast, the interface is neat, the learning curve is "flat"! What more could you want? (Point and Click here!)