This lesson discusses the notion of differentiability. Students gain practice in showing that a function is differentiable at a point and over an interval. Students also see how functions can fail to be differentiable. Students learn that differentiability implies continuity, but that the converse is not necessarily true. Piecewise functions are included to show that they may or may not be differentiable. Finally, students are shown the Weierstrass function--a function that is continuous everywhere, but differentiable nowhere.

This lesson is appropriate for both AP Calculus AB and BC.

Detailed answer keys are included.

The topics covered include:

-Differentiability at a Point

-Differentiability on an Interval

-One-Sided Derivatives

-Practice with Intervals of Differentiability

-Ways in Which a Function Can Fail to be Differentiable

-Differentiability and Continuity

-Differentiability of Piecewise Functions

-Continuous Everywhere, but Differentiable Nowhere

In this product, you will find the following:

-Powerpoint Presentation (24 slides)

-Guided Notes (4 pages, a "filled in" copy is also included)

-Do Now Slips (4 slips on one page, answers included)

-Exit Ticket Slips (4 slips on one page, answers included)

-Homework Assignment (2 pages, a detailed answer key is included)

This lesson is part of a larger unit on Derivatives. The complete unit includes the following:

-Rates of Change

-The Derivative Function

-Derivatives of Polynomials

-Graphing the Derivative

-Higher Derivatives

-Differentiability