Algebra Comments

Algebra Folder

Graphing

Alg 1 graphing (92)

This program was written to illustrate and explain basic concepts of linear equations for Algebra I students.  Due to the fact that we have no access to Macintosh computers, these pages are intended to be used as transparencies. 
-Graphing Lines
-Horizontal and Vertical Lines
-Distance Formula (Pythagorean Theorem)
Needs much improvement.  Would be suitable for Algebra I.

Reflections&Trans(93)calc

This program discusses reflections, translations and dilations of functions.  It seems fairly well done and I would like your feed back on it.

Sequences and Series

Arith sequence (94)

This program gives some examples of arithmetic sequences.
  
  Needs improvement.  Use in Algebra II or Pre-Calc.

ArithmeticMeans(94)

This program defines, gives examples and problems involving arithmetic means.  It does not use Mathematica.

Needs improvement.  Use in Algebra II or Pre-Calc.

Arithseq&series(94)

This program finds añ  and d of an arith. sequence.

Needs improvement.  Let Debra Woods know what you need.

Functions

Fcns and Inv (94)

This lesson allows the student to discover and use the Ordered Pair Test and the Vertical Line Test for functions.  The last part of the lesson deals with the graphing of the inverse.

- Functions and their inverses
- Function tests
- Inverses

This has a very good and detailed explanation of what a function is and what inverses are.  Highly recommended.  Let me know if you need additions, but this one is pretty complete.

GreatestIntegerFcn(92)

This program show pairs of points and graphs of the basic Greatest integer function.  It has great potential, and should possibly also include more examples and a thorough treatment of these functions.
I suggest improvement.  Suggestions?

Relations&Functions(93)

Righteous Relations and Friendly Functions

This program includes the following Subheadings:

Great DOMAIN  and Home on the RANGE

LINEAR natured RELATIONS

Funky FUNCTIONS

The program does a good job of explaining the concepts, however all of the code is hidden and I need to know if anyone wants it to show.
Also, I ran into a problem while running the cell under example 2.

Very Creative, worth checking out.

Linear Programming

Linear.Program(93)

This program includes history and motivation of the problem of linear programming.  It seems to be more of a tool for a student to sit down and use rather than for classroom demonstration.  

Look it over and give me some feedback on what you'd like changed for classroom use.

Solving equations

Linearsolve-GBeck

This notebook discusses the solutions of systems of equations  The built-in functions Solve and NSolve are the usual way to solve such systems.  The defined function WalkSolve solves a system step by step. 

This notebook was developed by George Beck at Wolfram Research, and is not officially part of our project.  Check it out anyway and see if there is anything usable that we can build on.

Lines

Lines(93)

This program begins by plotting more and more points of a two dimensional linear equation to show that a line is formed.  Then it discusses slope and animates the line using different slopes.  Lastly, it examines the y-intercept.
  In my opinion there is some good stuff here, and with some minor changes like more discussion, and some graph color, this could be really good.

Slope Intercept (92)

This program is worth looking over.  It discusses how to use Mathematica and then follows up with a discussion of slope and intercept.

Slope of Line (92)

This program needs a lot of work.  It begins by stating the definition of slope and then gives a few examples.  The last piece of executable code does not work.  If anyone sees any value in this program, let me know.

Lines&Parab(93)

This program has little explanation.  It begins by showing four intersecting lines.  I added color to help distinguish between the lines.  It then asks some questions about the lines, and proceeds to four parallel lines and questions about their equations.

The second part of the program plots parabolas and asks the same questions about the coefficients that most of the other parabola programs in these lessons include.

My recommendation is to dump this one.

PolynRoots(93)

This project provides two programs for teachers to use when discussing the roots of polynomial equations.  The first program simply graphs a series of polynomial equations varying by the constant factor,t.  These graphs may be animated while various aspects of the graphs are discussed.  The number of roots, number and location of real roots, and conjugate nature of imaginary roots
are some of the topics that could be discussed.
     The second group of programs graphs the same equations beside a graph of the roots of the equations on the real-imaginary axes.  This enables students to "see" more clearly the location and number of the real roots, the occurrence of double roots,and the location and the conjugate nature of imaginary roots.

There is not much discussion in this program, but the idea is good and the animations are good, although, I should go back and add some color.  Give me ideas on improving this one, I feel it has great potential.

Rates of Change(94)

This program begins by plotting points and showing the difference between variable rates of change, and constant rates of change.  Some of the code doesn't seem to work.  The program then falls into the same old what happens to the plot if you change the slope or y-intercept.  I think that we can possible incorporate parts of this into some of the other line programs.

Systems

Systems of Equations(92)

This project begins by using two sets of  raw data to produce a graph of points.  The lines of best fit are formed from these data. From these  data, the point of intersection and many other aspects of our graph are discussed.  The rest of the  unit focuses on solving equations algebraically in addition to the  graphing method.  Most of the problems presented are application problems.

SystemsofLinearIneq(92)

The following notebook allows you to graph various systems of inequalities and linear equations. Students should be able to recognize the difference between linear equation and linear inequalities and graph both.

Uni curve intersection

This program uses animation to demonstrate the possible intersection of various curves.  Among them are:  The intersection of a line and a circle, a line and a parabola, and two parabolas.  There is some room for improvement here in that other possible intersections can be included, and it could also act as a lead in to solving systems of non-linear equations.

Three Dim

3DimCoords(93)

This program provides a very basic introduction to three dimensionsal coordinate systems.  It begins by introducing the three dimensional coordinate axes and follows by animating a point moving along a path in three dimensions.  The animation also gives the coordinates of the point as it moves.

Plane Sections

This program begins by explaining how to draw pyramids and prisms.  It then shows the plane sections parallel to the bases of the 3-D pyramids and prisms.  Finally it animates the plane sections of the prism moving through the object.  It may be interesting although somewhat difficult to animate a plane section of the pyramid.

Planes(94)

This project shows the three possibilities for the intersections of three planes by drawing the planes, and animating a rotation to show several views.

RotatePlanes(93)

The purpose of this program is to  animate the rotation of three mutually orthogonal planes.

Uni3-D

This program introduces 3 dimensional space, ordered triples, and planes.  It then shows the possible intersections of 2 planes, and the intersection of 3 planes at a point.

Word Problems

Alg I work prob(92)

Author's description:
   Unity High School does not offer Calculus as part of its high school curriculum.  Therefore, I intend to give some examples of possible uses in Algebra I.  In the teaching 
of Algebra I at Unity High School, Mathematica can be used to supplement the material 
in the text in the following manner.  I have chosen problems at random to demonstrate Mathematica's value.  The text used to teach Algebra I at Unity High School is the 
Merrill Algebra I book.  Chapter 1 involves the math topics concerning Expressions and Equations.  Section 5 first introduces the distributive property and simplifying expressions.  The following problems are from the text.

Word Prob D=RT(92)

This program gives a fairly good and delightful approach to solving the very basic algebra I form of D=RT.

Word Problems(92)

This program is worth looking in to.  It demonstrates differnent stategies of solving some common word problems.

Calculus Comments

DE-Harm.Motion(93)

In this lesson, we will discuss one application of the use of second order linear differential equations with constant coefficients.  
										a y'' + b y' + c y = f(x)
The application of interest in this case is harmonic motion.  We will limit our discussion to free vibrations, that is, when f(x) = 0.

The focus is on using Euler's formula to transform the solution equation with imaginary exponents into the CiS form.

Area under curve (94)

Description:  This lesson graphs trapezoids under a parabola
 to compare areas.  It explains some Mathematica  language,also.  The commands used are explained.  This may be a good first program to look at for new users.  The program could use a little sprucing up.  I look forward to you comments.

Areas(94maxi)

Title:  Equal areas of irregular shapes
Description:  This lesson graphs areas between curves.  It finds
areas between curves, also.  The Mathematica  language is
discussed as well.

This program is somewhat of an extension of the Area under curve (94) program.  There are some bugs in the last part of this one.
Let me know if this is useful, or what improvements can be made.

Calculus graphing (92)

This file contains three lessons on graphing.  The first is titled Lesson 29 - translations.  The student is given a plot and asked to change the code to produce the translations.  I would suggest a possible change by plotting both the original untranslated plot and the translated plot together in different colors.
The second lesson is titled Lesson 39 Rational Functions II.  This lesson discusses x-intercepts and asymptotes.
The third lesson  is called Lesson 44, Factors of Polynomial Functions.  It covers the turning point theorem and the average value of roots.

Calculus Stuff (92)

This file contains lots of different calculus demonstrations.  There is not much text that explains anything, but the graphics are good and the animations are great.

The first animation is a saddle in 3-space.  The second is called limiting rectangles.  It animates the rectangle approximation of area under the curve y = Sqrt[x] as the number of rectangles increases.

The area under the curve demo allows the user to alter the picture of rectangles under the curve.  There is some impressive code in this section.  It is worth looking into.  Keep me informed as to whether it is easy for you to use.

The section titled Derivative stuff shows a curve and the tangent lines at various points and then shows the function f and f', f'', and f'''.  My only suggestion is to include some comments.

The last section is called Funky Fonts.  It does some probability (not using Mathematica and it uses funky font characters as variable symbols.

CalculusProblem(92)

Description:   This is a calculus problem concerning "on the  job" teaming.   As workers are teamed together, they tend to interfere with one another.  This program shows various ways to solve the problem.

ContourPlots(94)

This program plots a paraboloid and some of its level curves.  It then animates the level curves falling off of the paraboloid and on to the xy-plane.

Cycloids

There is no written explanation in this file.  However, there are two extremely nice animations of cycloids.  Worth looking at.  A very good visual explanation of how cycloids are generated.

Derivative of tangent(92)

This program uses the definition and graphing to find the derivative of the tangent function.  It shows the average growth for smaller and smaller values of h.  It might be nice to animate it someday.  Then the program shows that the derivative of Tan[x] is Sec¤[x].

DirectionFields(94)

This project is a demonstration to show the visual behavior of solutions of differential equations of the form dy/dx = f(x, y).  It creates a direction field, isoclines, and shows possible solutions to the differential equation.

Ellipsoids(93)

This program introduces the equation of an ellipsoid, the plot of an ellipsoid, it's cross sections and some exercises about ellipsoids.  In the section on cross sections, it also includes the three traces.

Equation of Tan line (92)

In this program,  examples of various curves are graphed along with their tangent lines at different points.  The tangent line is then animated moving along the curve.  The equation of the tangent line is also given for each point.  There are many different examples in this program, including some functions with infinite discontinuities.

Fcns&Derivs(93)

This program gives four examples of polynomial functions and their derivatives.  It then follows with some exercises for the students.  There is no color used in the graphics, and the function and its derivative are distinguished by dashing.  I look forward to suggestions on this one.

Limits(93)

Description:  This project is intended to be used near the beginning of a traditional course in introductory Calculus.  Because the limit is in a sense the glue that holds all of the calculus together, it is important that students have a clear understanding of this topic.  This project is designed to provide the student with an intuitive feel for the limit by presenting a few common functions along with their graphs and tables of function values.  Examples include functions whose limits do not exist for certain values of the independent variable.  Mathematica is used to calculate limits so that students may verify their intuitive guesses as to what the limit should be in each example and animation is used to provide a visual feel for what is taking place in the function values as the independent variable moves closer to a particular value from either direction.

Limits(94)

This program begins with a very interesting, and pretty example of using limits of the areas of polygons as the number of sides increase to show that the area is approaching the area of a circle.  The example is animated.

The program then proceeds with a rocket trajectory problem.   This has nothing to do with limits, and should probably be separated into its own program.

LocalCoord(94)

This code demonstrates the tangent, normal, binormal system (local coordinate system) as it moves along a circular helix 
r(t) = <2 cos t, 2 sin t, t¤/20>.   The path is animated.

NewtonsMethod(93)

This program demonstrates the use of Newton's Method.  Some of the problems with using Newton's Method are also addressed.  For future updates, I would suggest using color graphics.  The program makes use of animation.

P-series(92)

Description:  This lesson serves to utilize Mathematica's animation
 capabilities to assist students in visualizing the behavior of the classic 
 function 1/x∏ in the first quadrant as p ranges over natural numbers. 
 In particular, it focuses upon how:
       • {x,f(x)} = {1,1}, ∀  p       
       • f "hugs" the x axis as p increases
       • area under f is finite for p>1 and infinite for p=1, given an upper
        bound ∈ ( 0, \ ).
       Too, students will  in general become more familiar with inverse 
       function curves.

Polyno&Deriv(93)

The purpose of this lesson is to show the relationship between the first derivative and the graph of the polynomial function.  Mathematica  will allow us to plot the function, the derivative, and the slope of the graph at several discrete points.

RatesofChange(93)

This program examines average rate of change approaching instantaneous rate of change.  There is not much explanation, and improvements need to be made.

RatesofChange(94)

This program discusses the average rate of change of a function as it approaches instantaneous rate of change.  It animates the secant line on a function as it approaches the tangent line for smaller and smaller delta-x values.  Finally, it animates a tangent line moving along a curve.

SumofInfiniteGeomSer(94)

This program seems to be trying to show that some infinite geometric series do have finite sums.  The question is when and how?  The program doesn't explain that and I'm not sure the code works properly.  I would like to take this idea and rewrite the program from scratch.

TrapezoidRule(92)

Description:	
						The lesson successively draws the curve, shows the area, breaks the region into trapezoids, computes the area of one trapezoid, sums the areas of all the trapezoids and then computes	the area using NIntegrate.
						
						I would say we should try for a better color in the graphics.  This is an excellent program, worth looking at.

UniNewton's Method

This program should be merged with the Newton's method above which is much better than this.  This program is not finished.

Complex Comments

Complex Polygons

BRIEF DESCRIPTION
This Mathematica  notebook contains a library of polygons in the complex plane.  You will input a complex number z of your choosing and then pick a polygon from the library.  The notebook first displays the polygon, and then displays the polygon after it has been rotated through the argument of z and had its size dilated by the absolute value of z.  In otherwords, the points in the complex plane that make up the polygon are all multiplied by z.

Some of this notebook contains pictures that do not have the code that generated them.

Geom add complex(94)

Brief Description: Two non-zero complex numbers z1 and z2 are input.  Providing that zero, z1, z2, and z1 + z2 are all distinct, the graphs of these four numbers are the vertices of a parallelogram.

Geom mult complex(94)

Brief Description: A complex number z = x + iy can be written as z = r ( cos θ + i sin θ ).   The number r is the modulus of z, and the angle θ  is the argument  of z.  This notebook shows graphically that when two complex numbers are multiplied, their arguments are added.

Geometry Comments

GeometricDiscovery(93)

In beginning calculus it is common to show how to evaluate #ºıx¤dx directly from the definition of the definite integral by using the formula [n(n+1)(2n+1)]/6 for the sum of the squares of the first n integers, 1¤+2¤+3¤+...+n¤.  In most texts this formula is presented without consideration of its genesis.  The formula is usually proved using mathematical induction, but induction really does little to explain how the formula was discovered in the first place; in fact, students often object that there is a certain "circularity" to the process since they are being asked to prove something that they have no real reason to predict might be true. This project shows that there is a simple geometric technique for demonstrating the origin of the formula.  This technique can even be used by pre-algebra students to find the sum of the first n squares for any particular n even if they have not yet learned to use a variable.  For students who do understand variables, working with the this technique and Mathematica can provide a nice exercise in algebraic identities.

OverlappingTriangles(94)

This program only looks at the triangles in a picture that overlap.  The code to this program is not there, so modifications cannot be made.

Parallelogram Area(94)

This is a very good visual demonstration that graphically shows that the area of a parallelogram is base times height by animation.  The animation slides a triangular chunk from one side to fill the void on the other side of the parallelogram, thus forming a rectangle.

Polygons(94)

This code provides a visual demonstration of what a polygon looks like and how many diagonals can be drawn.

PolygonsToCircles(92)

Description:  This lesson investigates what happens as one allows the number of sides of a regular polygon to approach infinity.  This is similar to the limits program in the calculus section.

PythagoreanTheorem(92)

This program demonstrates the use of the pythagorean theorem to find the distance between two points.

Rotation,vol,SurfAr(93)

This program rotates an equilateral triangle about it's vertex and through animation shows the solid that is formed.

Conics Comments

Parabola Plotter(94 maxi)

This is a very good program that defines a parabola, shows graphically through animation that the focus and directrix are the same distance from each point on the parabola, and plots any parabola through input of the coefficients, a, b, and c .  Finally, the program will find the vertex, focus and directrix for a parabola, given the coefficients a, b, and c.

Conic Equations(93)

This notebook shows the graphs of circles, parabolas, ellipses, hyperbolas, and a rotated ellipse.  It is a good start, but could use some extra stuff.  For example, perhaps it should plot the directrix, and focus of the parabolas, maybe the foci of the hyperbolas, and ellipses.  Any other suggestions?

conic sections(94)

This notebook shows a right double cone being intersected by planes at various angles, thus showing the conic sections produced.

Conics(92)

Description:  This lesson treats conics as graphs of
a general second degree equation in two variables and relates each of them to their respective standard equation form through the concept of locus.  Each conic illustrated becomes an integral part of the design above.  Other treatment  of conics mentioned will be developed at a later time.

Some parts of this program are incomplete.

Ellipse(93)

This notebook demonstrates the definition of an ellipse through animation.  It is worth looking at.

EllipseMania(92)

This notebook provides a lesson for the student to learn about ellipses.  It doesn't explain much, but asks leading questions for the student.

GraphConics

This notebook is intended to give a working idea concerning three conic sections:  circle, parabola, and ellipse.  This notebook is well thought out and ties the ideas together.  Color needs to be added to the section on circles.

Parab roots

When studying the parabola, many students become confused about what is happening when the roots are no longer real roots.  This program will attempt to show that there are always two roots to the quadratic equation, whether the roots be real, one root of multiplicity two, or two imaginary roots.

Parabola definition

Previously we have seen that the function f(x) = a x¤ + b x + c generates a parabola.  The purpose of this lesson is to generate a parabola by using the definition of a parabola.  This lesson shows that the distance from the focus to a point on a parabola is equal to the distance from the point to the directrix by using animation.

Parabola2again(92)

This notebook is similar or nearly exactly the same as several other notebooks in this folder.  The lesson shows the effects of changing the coefficients of the quadratic equation.  This notebook may not be necessary to keep as it is similar to others.

parabolas(11/94)

This notebook shows the effects of changing the coefficients of a quadratic polynomial function of one independent variable.  Animation is used to give a visual demonstration of the effects.  Questions are asked at the end of the lesson to tie the ideas together.

Linear Programming

Linear.Program(93)

This notebook shows some examples of solving linear programming problems.
I found this program very confusing.  Please give me suggestions as to what would be useful.

LinearProg(94)

This notebook provides the steps one should take in solving a linear programming problem along with an example using the steps given.  Could be improved if the graphics are done in color.

Physics

Motion(94)

This miniproject determines the acceleration, the velocity, and the total time it takes for an object to move without friction down a straight ramp of a given length and a given inclination angle. It then uses straight ramp segments to approximate a parabolic ramp, and determines the approximate length of the curved ramp and the approximate time it takes the object to reach the bottom of the ramp.

Ramp(94)

Five Examples of time for a frictionless ball to roll down a slanted ramp are given.  The first example uses easily verifiable numbers, all others use a one unit tall and two unit wide ramp.  Examples 1 - 4 are straight lines and the curved result of Example 5 is counterintuitve.

Traject2(92)

Description:        This lesson is intended for a high school physics class demonstration/lab after the students have been exposed to projectile motion in the usual treatment.  Takes into account the effect of air resistance in the trajectory.

Pre-Calculus

Graphingwithparam(93)

This notebook provides a good explanation on the use of parameters in graphing.  The author of the code states:
     Personally, the use of parameters is one of the subjects that I do not emphasize enough.  I know that I expect my students to deal with this subject when they are enrolled in my Physics course, however, parameters are not often covered in my Math courses.  Hence, I intend to use this notebook to help bridge the gap.

LimaconsAndCardioids(92)

This notebook provides an indepth discussion of Limacons and Cardiods.  It includes animation showing how to draw the various shapes and a discussion on generating the plots.

MorePolar3(92)

This is a notebook that contains various non-traditional polar plots.  It may provide a student with the motivation to try something creative in polar plotting.

OtherPolar4(92)

Roses, limaçons, and cardioids are examples of graphs whose equations look particularly nice in polar coordinates.  We will now look at some other graphs whose equations also have this property.  We will specifically not consider lines, parabolas, ellipses, and hyperbolas at this point since they tend to be handled more easily in rectangular coordinates.

Polar Intersections

This notebook provides a demonstration of the possible intersections of polar graphs.  Animation is used to demonstrate why certain point(s) of intersection may not be solved for algebraically.

Polar5 (92)

The actual title of this notebook is "Experimenting With Polar Graphs".  The notebook gives several examples of non-traditional polar graphs, and then encourages the student to continue to experiment.

PolarCoordSystem1(92)

This is the first notebook in a series of lessons on Polar graphing.  This lesson explains the basics of a polar plot.

PolarGraphs(93)

This notebook contains a variety of polar graphing concepts.  It begins with plotting polar points, however, not much in the way of a written explanation is given.  It then continues to plot some of the traditional polar plots, and includes an example of the intersection of polar plots.

PolarPlots(93)

The object of this lesson is to teach students to analyze
basic polar graphs through experimentation with 
computer graphing.  This notebook only contains examples of rose type graphs, but it encourages the student to try different n-values to adjust the number of petals.

Polyn&dominance(93)

This project will develop three lessons which will investigate and compare the graphs of different functions. The objective of the first lesson is to help the students become familiar with the properties of odd and even functions and also to see what happens when a function is negative. The second lesson takes polynomial functions of the same degree and compares their graphs as x gets very large. The third lesson looks at the growth of the square root function, the sin function, the natural log function, a polynomial function, and last of all the exponential function. Students will be given an "Investigation Sheet" on which to record the discoveries they make on the computer.

Rose Curves(92)

Description: The following notebook allows students to explore 
Rose Curves by changing constants in the equations:          
r = a Sin(nt), r = a Cos(nt).  After running the animations and
 looking at the graphs, the students should be able to generalize 
 how the constants change the graph of the curve.
 
 This notebook is a complete treatment of roses, including, stretching, rotating, and translating.

Statistics

DataAnalysis

This is a data set of actual data from XXXXXX High School.  Theses are student admission scores from students selected for admission from the pool of applicants in Spring 1990.  This program provides an analysis of the predicted GPA's of these students.

Descriptive Stat

Brief Description
     This project is an illustration of the built-in functions of the Descriptive Statistics package.  This package is automatically loaded when most other statistical packages are used.
     This project also provides an explaination of the measures used in descriptive statistics.

Regression

This project takes an arbitrary set of bivariate data (paired data) and looks at vaious ways to "fit" a model to the data.  Models which were fitted include linear, quadratic, cubic, trigonometric, logrithmic, and a sum of non-integral-power ploynomial-like functions.
					Part 1 of this project uses the "Fit" command and allows the user to compare the resulting models on the scatter plot visually.  The statistical packages are not used in this section of the project.
					Part 2 of this project uses the Linear Regression Package and the "Regress" command to obtain a statistical analysis of the models, and uses this analysis to compare three different models. 
					Part 3 of this project uses the NonLinear Regression Package to obtain several non-linear models and alter the method used for the fit.

Trigonometry

Dom.,Range(trig)(92)

Description:  This lesson serves to utilize Mathematica's graphing capabilities to help students visualize the results of changing a function's domain and range in various ways.  The intended audience spans over several years, from perhaps middle school to undergraduate mathematics.  This is not a formal treatment of domain and range concepts, since they are only treated here as intervals.

generateSinefromUnitCir(93)

Description:    This project introduces parametric equations showing
their relationship to the unit circle and the Sine and Cosine functions. 
Code shows an excellent animation of how the sine plot relates to the unit circle.

Graphing Sine(94)

This is an excellent program that makes use of animation to demonstrate the effects of the coefficients in determining amplitude, period and phase shift.  Worth looking into.

PolarGraphs(94)

This notbook contains a demonstration to show to the students.  It will show why a graph of two polar equations may have intersections that are not solutions to the system of equations.  The example here is:
  r = sin 2t
  r = 1

Simpson,C-Sinusoids(92)

This notebook investigates the effects of the coefficients of a sine function on amplitude, period and phase shift.  It also animates a sine and cosine function as they relate to the unit circle.

SinCosTan(93)

This notebook looks at the graphs of the sine, cosine, and tangent functions and their properties such as amplitude, period, and phase shift.  The last part of the notebook looks at combinations of functions such as y = sin x + cos x.

SSA

This notebook provides a graphical animation of the three ambiguous cases of the Law of Sines.

Trig Fcns(94)

The subject matter of TRIGONOMETRY is based upon six trigonometric functions.  In this section we will introduce the first three--the sine, cosine and tangent functions (abbreviated sin, cos and tan).  
This program uses animation to relate the functions to the unit circle.  In my opinion, this is the better of the notebooks in this section that demonstrate that point.

TrigParameters(92)

Description:  This project will show how different parameters effect the sine curve.  The parameters will appear in the form of f[x]= a Sin[nx+b]+c ,where a,n,b,c are the parameters we will change individually.

This is just more of the same.

trigsum

The purpose of this lesson is to investigate the sum of sine and cosine functions each with the same period, and to derive the relationship between this sum and a single sine function.

Waves-sin and cos(92)

Description:
               This notebook is a look at the Sine and Cosine waves and the parameters that determine the shape of the graphs.

Word Problems

RetirementApp(93)

This is an example of a retirement program to motivate students to look into their financial futures.