EMAT 6690
Spring 2000
An Instructional Unit on
the Derivative
Michael E. McCallum
This is an instructional unit designed to teach the derivative to a college freshman. The unit is organized into nine one hour lessons over a two-week period. The unit will begin with concepts from basic physics as a motivation for needing the derivative. Using these concepts we will develop the difference equation and explore its meaning. We will use Geometer Sketch Pad to demonstrate the effect of the secant line becoming the tangent line as the distance between two points in a function approaches zero. Finally we will introduce the derivative in its form as the limit of the difference equation as the denominator approaches zero.
Assumptions
It is assumed that the student has just completed a unit on limits and is familiar with the physical concepts of time, distance, velocity and acceleration. Also, it is assumed that the student is already familiar with the basic analytic geometry of functions in one variable up to logarithmic functions.
Day One: Rates of Change
Review the concept of rate of change with the students. Begin with the example of acceleration, velocity, distance, and time and develop the relationships between these. Next, discuss mass, force, and work and show how they are related to acceleration, velocity, distance, and time.
Acceleration: a (t ) = initial acceleration+ coefficient of acceleration x t
Velocity: v (t ) = initial velocity+ acceleration x t = initial velocity +ta(t )
Distance: d (t ) = initial distance+ velocity x t = initial distance + tv(t )
Mass: A physical term indicating the quantity of matter contained in an object. Mass is often confused with weight because the terms used to describe mass and weight are the same. If an object is said to have a mass of 1 kilogram, what is meant that the object would weigh 1 kilogram if a constant acceleration of1 unit of gravitational force were applied to it constantly. That is, if the object were at rest on the surface of our planet. This object would have a mass of 1 kilogram, no matter where is located, even in outer space.
Force: f (t ) = mass x acceleration= mass x a (t ).
Energy: E (t ) = force x velocity= f (t ) x v (t )
Work: W (t ) = force x distance= f (t ) x d (t ).
Click here for Lesson Plan.
Day Two: Average Rates of Change.
Develop the difference equation using the concept of average rates of change.
Build upon the previous day's lesson using the physical concepts used to develop the difference equation and then show how it can be used to calculate average rates of change over a period of time.
Click here for Lesson Plan.
Day Three: Instantaneous Rates of Change
Introduce the interpretation of the difference equation as the secant line. Demonstrate this interpretation using Geometer's Sketch Pad (GSP) using several examples of differentiable functions. Show what happens as the secant line becomes a tangent line. Show that the slope of the tangent line represents the instantaneous rate of change in f (x ) at the point of tangency. Show that as the denominator of the difference equation approaches zero, the result is the instantaneous rate of change at a point.
Useful GSP Sketches:
Click on the titles below to observe examples of secant lines generated by the difference equation for various functions. These are GSP sketches that can be used to demonstrate the concepts referred to above.
Click here for Lesson Plan.
Day Four: Introduce the Derivative
Introduce the derivative of a function at a specific value for the independent variable as the limit of the difference equation as the denominator approaches zero. Revisit examples from day three using the definition. Use GSP to illustrate the concept of the derivative. Click here for examples.
Click here for Lesson Plan.
Day Five: Operations on Derivatives; Derivatives of Polynomials
Demonstrate the linear nature of the derivative. Show how the linear nature of the derivative allows us to do operations on derivatives. Use the additive property of derivatives to show how to find the derivative of a polynomial.
Click here for Lesson Plan.
Day Six: The Product Rule.
Introduce the product rule. Show how the product rule is much simpler to use to calculate derivatives than using the difference quotient method.
Click here for Lesson Plan.
Day Seven: The Quotient Rule; Derivatives of Rational Functions.
Introduce the quotient rule. Show how to find the derivatives of rational functions using the quotient rule.