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The Concept of a Derivative
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| | The Derivative As Rate of Change: A Numerical Approach
This tutorial introduces the derivative with numerical examples leading up to the (informal) limit definition. Three of the four examples are based on real situations changing over time. The fourth shows how to calculate difference quotients numerically on a calculator. The forward difference quotient and the symmetric (or balanced) difference quotient (nDeriv) are carefully explained.
 The Derivative As Rate of Change: A Numerical Approach
The Definition of Differentiation
This one-page discussion of the definition of the derivative from the analytic point of view includes links to two animations and can help you introduce the definition of derivative to your class. It is part of a Web site that includes a full outline of calculus. The discussion is thorough, complete, correct, and short.
The Definition of Differentiation
Army Versus Navy: The Long Trip
This interactive Web site features a Java applet dynamically showing the relationship between secants and tangents, average rate of change and instantaneous rate of change, and average velocity and instantaneous velocity.
Army Versus Navy: The Long Trip
Continuity and Differentiability
This page gives a brief but very good discussion of continuity and differentiability. It begins with the definition of continuity and then discusses the types of discontinuities (jump, vertical asymptotes, and holes in the graph). This leads to a discussion of the fact that some continuous functions are not differentiable.
Continuity and Differentiability
Derivatives
This page is part of a site maintained by Sandra Halfacre of White Station High School in Memphis, Tennessee. It contains links to tutorials, slide shows, animations, and applets helping students to understand the concept of the derivative, the relationship between the graphs of f and f ', and derivative rules.
Derivatives
Average Rate of Change
This Web site, part of a calculus tutorial, contains a good, clear explanation of the average rate of change of a function. It explains that this concept is the same as the difference quotient. The graphics are clear, but they are not animated. The explanation includes a sentence with the units for the average rate of change. An example with units of tons and years is followed by a problem for the student to solve involving meals and days where the data is given in a table.
Average Rate of Change
Introduction to the Derivative
This page gives a very complete summary of the main points related to the introduction of the derivative. There is an excellent set of notes in which each main idea is illustrated with an example. Links at the top will take you directly to the topic you choose. The page also links to 20 interactive true/false questions and six interactive review questions on derivatives. There are also links to utilities.
Introduction to the Derivative
The Graph of a Function Versus the Graph of Its Derivative
This brief tutorial presents the relationship between the graph of a function and its derivative, as well as increasing and decreasing functions and local and absolute maximums and minimums. Its goals are for the student to recognize graphically when a function is increasing or decreasing by looking at the graph of its derivative and to recognize graphically the local maximum and the local minimum.
The Graph of a Function Versus the Graph of Its Derivative
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