Exploring Differentiability and Continuity in Calculus

Exploring Differentiability and Continuity in Calculus

Learning Objective

At the end of the lesson, you will be able to identify if a given graph is differentiable or continuous.

What do you already know about differentiability and continuity?

Understanding Differentiability

Differentiability refers to the existence of the derivative of a function at a given point.

Properties of Differentiable Functions

Differentiable functions are also continuous, but the converse may not hold true.

Identifying Differentiable Functions

To identify if a function is differentiable at a point, we can use the definition of the derivative.

Understanding Continuity

A function is continuous at a point if the limit of the function as x approaches that point exists and is equal to the value of the function at that point.

Types of Discontinuities

There are three main types of discontinuities: removable, jump, and infinite discontinuities.

Testing for Continuity

We can test for continuity using limit properties and algebraic manipulations.

Differentiability vs. Continuity

All differentiable functions are continuous, but not all continuous functions are differentiable.

Graphs of Differentiable Functions

Differentiable functions have smooth, non-cornered graphs without breaks or jumps.

Graphs of Continuous Functions

Continuous functions have graphs that can be drawn without lifting the pencil and have no holes or jumps.

Analyzing Graphs for Differentiability

We can analyze the sharpness of corners and presence of breaks to determine differentiability.

Analyzing Graphs for Continuity

Look for any breaks, jumps, or holes in the graph to determine continuity.

Interactive Exercise: Identifying Differentiability

Present students with graphs and ask them to determine if the functions are differentiable at specific points.

Interactive Exercise: Testing for Continuity

Engage students in evaluating the continuity of functions at various points using limit properties and algebraic manipulations.

Wrap Up: Comparing Differentiability and Continuity

Summarize the key differences between differentiability and continuity and their implications in analyzing functions.

Wrap Up Activity: Graph Analysis Challenge

Provide a set of graphs for students to analyze and determine the differentiability and continuity of the functions represented.

Write down 3 things you learned in this lesson.

Have students enter three things they learned in this lesson. With this they can indicate their own learning efficiency of this lesson.

Write down 2 things you want to know more about.

Here, students enter two things they would like to know more about. This not only increases involvement, but also gives them more ownership.

Ask 1 question about something you haven't quite understood yet.

The students indicate here (in question form) with which part of the material they still have difficulty. For the teacher, this not only provides insight into the extent to which the students understand/master the material, but also a good starting point for the next lesson.