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Differential Equations and Mathematical Modeling
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| | Introduction to Differential Equations
This Web site is a noninteractive tutorial providing an excellent introduction to a differential equations unit in either Calculus AB or Calculus BC. It begins with a word problem involving the spread of a rumor to motivate the concept of a dfferential equation. The ideas of separation of variables and initial conditions are established and well motivated. Graphs are provided.
 Introduction to Differential Equations
Introducing a Differential Equation
This is the first part of a three-page introduction to differential equations. Starting with the differential equation dy/dx = y, the important terms and concepts related to differential equations are presented, including the idea of a solution (but not any methods of solution), units, and an initial value. Several examples are worked in detail to explain the terms and ideas.
Introducing a Differential Equation
Differential Equations and Mathematical Modeling
This is part of a site maintained by Sandra Halfacre of White Station High School in Memphis, Tennessee. This particular page contains links to tutorials, slide shows, animations, and applets geared to help students understand the topic of differential equations, including antidifferentiation techniques, exponential growth and decay, and slope fields.
Differential Equations and Mathematical Modeling
Qualitative Techniques: Slope Fields
This is one of the 2,500 pages to browse through at sosmath.com, a site that has been maintained by Mohamed A. Khamsi and Helmut Knaust since 1995. This page has an excellent introduction to the idea of a slope field. Good use is made of colors in showing a curve and little tangent line segments at various points.
Qualitative Techniques: Slope Fields
Slope Fields (PDF)
This 23-page tutorial on slope fields is in PDF (Adobe Acrobat Reader) format. The tutorial is authored by Sean F. Ellermeyer, a professor at Kennesaw State University, who does an outstanding job introducing the idea of slope fields, pitched at a level appropriate for most AP students. You may want to direct your students to the URL to supplement the introduction you've given in class.
Slope Fields (PDF)
Qualitative Ideas and Direction Fields
This is the first part of a four-page discussion of slope fields (also called direction fields). The first page is a nice introduction that could be of use in introducing the topic with either Calculus AB or BC students. The discussion is limited to differential equations of the type dy/dx = ky. The presentation is quite good and includes provocative leading questions.
Qualitative Ideas and Direction Fields
Slope Fields
After a very short explanation of what a slope field is, this Web page links to two interactive slope field applets that will let you investigate a slope field in various parts of the plane. The functions are built in and cannot be changed. Very clear step-by-step, illustrated examples for using the TI-86 graphing calculator's built-in slope field program are included as part of the tutorial.
Slope Fields
Euler's Method
This presentation of Euler's Method uses the idea of the local linear approximations (the tangent line) to approximate the solution of an initial value problem.
Euler's Method
Population Growth Models
This Web site introduces different population growth models. The site is arranged in five chapters, starting with an introduction and a brief chapter dealing with the dP⁄dt = k P situation. Two more substantial chapters follow: World Population Growth (five sections) and Logistic Growth (six sections).
Population Growth Models
Modeling Population Growth
This Web site, part of the University of Minnesota Calculus Initiative, contains six components, some of which are interactive. The first few pages are instructive tutorials that examine simple differential equations modeling population growth.
Modeling Population Growth
MathServ DE Toolkit
The MathServ DE Toolkit is an online resource for solving differential equations. The user enters the equation and clicks, and the solution is returned. Twelve of the simplest types of differential equations are available from boxes on this page, the first being separable differential equations -- the only kind required for the AP Calculus Exams. Solvers for additional types of equations are available.
MathServ DE Toolkit
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