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Calculus: Early Transcendentals, Fifth Edition
By James Stewart
Michael Penna
Indiana University - Purdue University, Indianapolis
This page contains links to computer projects based on Mathematica 5 which accompany Calculus, Fifth Edition, by James Stewart.
The objectives of these projects are discussed here, and some suggestions for their usage are discussed here.
The projects in this section are background for material to be covered later.
Assigning the first and second projects is essential. Remaining projects can be done at the beginning of the term or postponed until they are needed.
It is worth working through the first and second projects in class early in the semester (on, for example, a computer whose display can be projected onto a wall screen) just prior to having students
do them. Working through them in class is a good opportunity for discussing details (such as how to get Mathematica up and running, and editing) which students without much computer experience will
probably appreciate. This is also a good way to make constructive use of time during which students are normally settling in to a new term.
- "Why Worry About Proving Things?"
This project is aimed at motivating the need for proof in Calculus. It encourages experimentation, and it involves some elementary Mathematica programming.
- Variations in the Graph of a Function
- Variations in the Graph of the Sine Function
These projects review how shifting and scaling affect the graph of a function. The first project uses a polynomial function, and the second uses the sine function.
- Amplitude, Period, and Phase Shift
This is a quick review of the amplitude, period, and phase shift of f(x) = A sin (Bx+C).
- Exponentials and Logarithms
This is a very brief review of pre-Calculus exponentials and logarithms in preparation for further development in Calculus: it emphasizes that prior to Calculus (ie, without introducing limits),
exponentials are defined only for rational exponents. This project does not use Mathematica: all computations can be done by hand.
- Linear Interpolation
This project is for the benefit of students who need to know something about linear interpolation: many pre-Calculus texts and courses do not cover this topic. This project does not use Mathematica: all
computations can be done by hand.
- Predicting the Slope of a Tangent Line
These projects have the student predict the slope of the tangent line to a function at a point by examining the slopes of secant lines. The first project computes limits as h approaches 0, and the
second as x approaches a.
- Linear Approximation
This project is aimed at applying the derivative and tangent line early (before discussing differentials).
- Guessing Limits Numerically
In this project numerical data is created, and from it a limit is to be guessed.
- The Definition of Limit
This project is aimed at showing that deltas exist for given epsilons.
- The Intermediate Value Theorem
This is a simple Intermediate Value Theorem project.
- The Intermediate Value Theorem and Graphing
- The Intermediate Value Theorem and Graphing
These projects discuss the Intermediate Value Theorem in the context of pre-Calculus graphing, and they also lay the foundation for the later discussion of graphing. The first of these projects
does not involve vertical asymptotes but the second does.
- The Bisection Method
The Bisection Method is a consequence of the Intermediate Value Theorem, and an alternative to Newton's Method for solving equations. The Bisection Method doesn't converge as fast as Newton's
Method, but it's very easy to program and it illustrates limiting behavior.
- Differentiation
The student is led through the computation of a derivative using the definition of derivative, and then asked to correlate the first three derivatives of a single function.
- The First Derivative and Slope
- The First Derivative and Slope
In the first of these projects, the student works through the computation of a derivative using the definition of derivative, and then is asked to correlate the graph of a function with the graph
of its derivative. In the second, the student is asked only to correlate the graph of a function with the graph of its derivative.
- The Graph of a Function and the Graph of its Derivative
The graphs of functions are given and the student is asked to draw the graph of their derivatives by hand.
- The Graph of a Derivative
- The Graph of a Derivative
Given the graph of a derivative and an initial value, the student is asked to graph the original function.
There are three different versions of several of the projects in this chapter: in each case one version involves a polynomial function, one involves a rational function, and one involves an
algebraic function. These different versions differ primarily in the Mathematica code, and are presented to accommodate different levels of courses, different levels of student abilities, and
the instructor substituting a different function, interval, etc, for the ones used.
Some of these projects require solving equations. Where the NSolve command is clearly better to use than the Solve command, it has been used. However, the instructor might also wish to have students
use it elsehwere and to restrict the number of displayed digits. (See A Brief Mathematica Tutorial: Part 1, Arithmetic and Simple Algebra and Mathematica's
Help for more details.)
- Maximum and Minimum Values on a Closed Interval
- Maximum and Minimum Values on a Closed Interval
- Maximum and Minimum Values on a Closed Interval
The student is led through the process of finding the maximum and minimum values of a function on a closed interval. The difference between these projects is that the first addresses a polynomial
function, the second a rational function, and the third an algebraic function with a fractional exponent.
- The Mean Value Theorem
- The Mean Value Theorem
These address the Mean Value Theorem.
- Finding Relative Extrema on an Open Interval: The First Derivative Test
- Finding Relative Extrema on an Open Interval: The First Derivative Test
- Finding Relative Extrema on an Open Interval: The First Derivative Test
The student is led through the process of finding the relative extrema of a function on an open interval using the First Derivative Test. The difference between these is that the first addresses a
polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.
- The Second Derivative and Concavity
This establishes the connection between the second derivative and concavity.
- Finding Relative Extrema on an Open Interval: The Second Derivative Test
- Finding Relative Extrema on an Open Interval: The Second Derivative Test
- Finding Relative Extrema on an Open Interval: The Second Derivative Test
The student is led through the process of finding the relative extrema of a function on an open interval using the Second Derivative Test. The difference between these is that the first addresses a
polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.
- Concavity and Inflection Points
- Concavity and Inflection Points
- Concavity and Inflection Points
The student is led through a discussion of the concavity and inflection points of a function on an open interval. These projects anticipate a more complete coverage of graphing. The difference
between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.
- Critical Numbers and Inflection Points
- Critical Numbers and Inflection Points
- Critical Numbers and Inflection Points
These projects have the student graph some functions with Mathematica and then label the critical points and the inflection points. These projects again anticipate a more complete coverage of graphing.
The difference between these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.
- Graphing Functions
- Graphing Functions
- Graphing Functions
These projects each give a discussion and example of qualitative graphing, and then, given a blank graphing grid and data about a specific function, ask the student to graph. The difference between
these is that the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.
- Graphing Functions Again
- Graphing Functions Again
- Graphing Functions Again
These projects are a follow-up to the previous projects: they just give a blank graphing grid and data about a specific function, and ask the student to graph. The difference between these is that
the first addresses a polynomial function, the second a rational function, and the third an algebraic function with a fractional exponent.
- Newton's Method: Part 1
- Newton's Method: Part 2
- Newton's Method
These projects present a basic discussion of Newton's Method. A couple of approaches are possible: the first of these projects alone can be assigned, the first two of these projects can be assigned
(for a more in-depth treatment), or just the third project (which attempts to compress the first two can be assigned.
- Ballistics
- Ballistics
- Ballistics
- Ballistics
These are four variations on the classic problem of describing the motion of an object that is projected into the air. The first and second of these projects use English units, and the later two
use metric units. The first and third specify the Mathematica code to be used, and the second and fourth leave writing the Mathematica code to the student. (Some students appreciate the freedom to do their own
thing.)
- The Iteration Method
This project presents a discussion of the Iteration Method.
- Integration: Riemann Sums
- Integration: Riemann Sums
These projects represent three different approaches to integration using Riemann sums. The second uses Mathematica's Student package, and the third asks the student to write some Mathematica code.
- The Fundamental Theorem of Calculus
This project discusses the Fundamental Theorem of Calculus, and emphasizes the definition of a function as an integral.
- The Fundamental Theorem of Calculus
This project describes an alternate proof of the Fundamental Theorem. It is a little off the beaten path, but it does anticipate Euler's Method and telescoping sums. This project does not use
Mathematica: all computations can be done by hand.
- Areas of Type I and Type II Regions
This is a basic "find the area" projects.
- Surfaces of Revolution
- Surfaces of Revolution
- Surfaces of Revolution
- Surfaces of Revolution
These projects are concerned with producing graphics that can be used to get a better feel for surfaces of revolution: students are not expected to understand the Mathematica code, although they should
be able to modify it to draw surfaces that might be of interest to them. In the first project the student is led through revolving the graph of y = f(x) about the x- and y-axes, and around the
lines x = k and y = k; in the second the student is led through revolving the graph of y = f(x) just about the x- and y-axes; in the third the student is led through revolving the graph of y = f(x)
about the x- and y-axes, and around the lines x = k and y = k, and then asked to revolve the graph of a different function about the same axes; in the fourth the student is led through revolving
the graph of y = f(x) about just the x- and y-axes, and then asked to revolve the graph of a different function about the same axes. (These variations are aimed at accommodating both class
schedules and student abilities.)
- Force and Work
One of the major applications of Calculus is to problems that involve force and work. This project is preparation for this application, and is aimed particularly at students who have not yet studied
Physics. This project does not use Mathematica: all computations can be done by hand. (This project repeats a small part of the work on variation and proportion related to Hooke's Law covered in Variation and Proportion so that that project does not necessarily have to be covered.)
- The Average Value of a Function
This project discusses the average value of a function, and illustrates its geometric significance.
- Approximate Integration
In this project we compare approximate integration by Riemann sums, the Trapezoidal Rule, and Simpson's Rule.
In the first project on Parameterized Curves students are encouraged to use their creativity to draw something interesting. To make things a little more interesting for the class as a whole, you
might want to consider offering a small prize - such as a candy bar - for the "best" offering.
- Parametrized Curves
- Parametrized Curves
In these projects the student is led through the graphing of several parameterized curves in the plane. In the first project, the code is provided; in the second project, the code is provided for
one curve, and the student is then expected to write the code for the others.
- Ballistics
- Ballistics
- Ballistics
- Ballistics
In these projects the student investigates the path of a projectile in xy-coordinates. (Earlier they were investigated in ts-coordinates.) The first and second of these projects uses English units,
and the later two use metric units. The first and third specifies the Mathematica code to be used, and the second and fourth leave writing the Mathematica code to the student.
- Polar Curves in Cartesian Coordinates
This project illustrates how to draw polar curves in Cartesian coordinates using Mathematica.
- Cardioids and Limacons
This project asks the student to draw several cardioids and limacons in the same graphic. The objective is some insight into the geometric significance of the constants a and b that specify curves
whose polar equations are of the form r = a + b sin theta and r = a + b cos theta.
- Lissajoux Curves
This is an investigation of Lissajoux Curves.
- Sequences and Series: Part 1
- Sequences and Series: Part 2
The first of these projects has the student define two procedures - one for generating some values for, and another for generating some graphics for, a user-specified series - and then apply them
to an analysis of the series. The second project asks the student to repeat the first project using some different series.
- Maclaurin and Taylor Series
In this project the student creates and graphs some truncated Maclaurin and Taylor series expansions.
- The Second Derivative Test Again
This views the Second Derivative Test from the standpoint of series expansions, and extends the Second Derivative Test.
- 3-Dimensional Coordinate Systems
This project has the student draw several basic geometric figures in space.
- Cylinders and Quadric Surfaces
- Cylinders and Quadric Surfaces
These projects have the student drawing and identifying cylinders and quadric surfaces in space. The first specifies the Mathematica code to be used, and the second leaves the writing of part of the
Mathematica code to the student. Some parts of these projects can be omitted if there seem to be too many.
- Quadric Surfaces
In this project the student is asked to identify the graphs of several quadric surfaces from implicit graphs of their second order quadratic equations in x, y, and z.
- Surfaces of Revolution
In this project the student uses implicit equations to graph some surfaces of revolution.
- Graphs of Functions of Two Variables
In this project the student graphs functions of two variables.
- Tangent Planes
In this project we discuss the similarities and differences between computing the equation of the tangent plane to the graph of a function and the equation of the tangent plane to the graph of a
level surface.
- Function Optimization
- Function Optimization
These projects are aimed at clearly identifying the process of finding the relative extrema for a function of two variables over an open set. The first project specifies the Mathematica code to be used,
and the second leaves writing the Mathematica code to the student.
- Solving Constrained Optimization Equations
This project illustrates how to use Mathematica to solve the systems of equations that arise in constrained optimization problems.
- Constrained Optimization
This project illustrates the geometry behind Lagrange Multipliers.
- Lagrange Multipliers
This project illustrates how to solve Lagrange Multiplier problems analytically, and the geometry behind Lagrange Multipliers.
- Lagrange Multipliers
This project illustrates how to solve Lagrange Multiplier problems analytically, and includes several more for the student to do.
- Vector Fields in the Plane and in Space
This project illustrates how Mathematica can be used to define vector fields and draw them in the plane and in space.
- The Gradient
- The Gradient
In these projects the student is asked to draw the graph of a function f = f(x,y), a contour plot of f, and the gradient field of f, and then to relate the graphics. The difference between these
projects is the function f.
- Integral Curves
This project asks the student to draw some integral curves of a vector field. It also relates the integral curves of a conservative field to the level curves of the potential function.
- Divergence
- Divergence
In these projects the student is asked to work with the divergence of some simple vector fields. The emphasis is on understanding what the divergence represents.
- Line Integrals in the Plane
This projects discusses the computation of line integrals in the plane.
- Line Integrals in the Plane, and Work
- Line Integrals in the Plane, and Work
These project discuss the computation of line integrals in the plane, and emphasizes their interpretation in terms of work. In the first project the student computes the work integral, and in the
second they are expected to guess whether it is positive, negative, or zero.
- Drawing Surfaces
This project illustrates how to draw parametric surfaces in Mathematica, and the strengths and weaknesses of drawing surfaces implicitly, as graphs of functions, and parametrically.
- Parameterized Surfaces
This project illustrates how to draw parametric surfaces in Mathematica.
- A Comparison of Approaches to Drawing Surfaces
In this project we compare the graphs of a cone obtained through implicit, function, and parametric plotting.
- Different Strokes ...
This project requires student to write code in Mathematica. In this project we illustrate that different types of information can come from viewing surfaces that have been graphed using different
parameterizations.
- Surface Area and Surface Integrals
The goal of this project is to clarify the steps taken in the computation of surface area and surface integrals.
- Spring-Mass Damping
Students are directed to draw and analyze the spriong-mass damping equations.
COPYRIGHT 2005 Brooks/Cole, a division of Thomson Learning, Inc.
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