Lesson #21
Definition of the Derivative

Quote of the Day:
"Discovery lessons, students writing to learn mathematics, the teaching of so-called general problem solving concepts, field trips, math lab lessons, alternate assessments, collaborative partner tests, student presentations, and open-ended problems should all be used sparingly.   I use some of them, but they have limited value.   Pencil-and-paper analytic solutions are the heart of mathematics education."
--Michael Stueben in Twenty Years Before the Blackboard

Objectives:

The student will apply the definition of the derivative to finding derivatives of algebraic functions.

Given the graph of a function, the student will determine the graph of the derivative.

1. Bellringer.

2. To introduce derivatives:




3. Notations for the derivative:


4. Examples:



5. Click here for the lyrics of the Derivative Song

6. Click here for a song about the Definition of the Derivative

8. Existence of derivatives:
The derivative does not exist at any of the following:



In the examples above, think of derivative as the slope of the curve at a point.

In the corner or cusp, the slope cannot be equal to two different values at the same point.

In the vertical tangent, the slope cannot be equal to infinity.

In the point of discontinuity, the slope cannot be equal to two different values at the same x-value.

If f(x) is differentiable,
Then it is also continuous.

The converse of this statement is NOT true:
Continuity does not imply differentiability.

9. Graph of the derivative:



Use tangents to the graph to approximate the slope at various points.   Then graph the values of those slopes on an axis below the original function.

10. Assignment: p. 187 (5, 7, 9, 11, 15, 21, 23, 25)

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Send any comments or questions to: David Pleacher