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Continuity and Differentiability (Topics 2.3-2.4)

Unit 2 - Day 5

​Learning Objectives​
  • Estimate the derivative at a point using graphs or tables

  • Explain the relationship between differentiability and continuity

  • Justify how a continuous function may fail to be differentiable at a point in its domain

​Success Criteria
  • I can compare derivatives at various points by estimating from a graph

  • I can find points of discontinuity and know that a function is not differentiable there

  • I can use limits to show that a function is not differentiable at a specific point

Quick Lesson Plan
Activity: Is This Rollercoaster Safe to Ride?



Lesson Handout

Answer Key


Most students have enjoyed a ride on a rollercoaster and have already associated smoothness with safety and steepness with thrills. Now they will connect these concepts to continuity and differentiability. They will determine relative derivative values from graphs and then formalize the connection between discontinuity and non-differentiability. Additionally, students will need to use the limit definition of a derivative to justify non-differentiability of continuous functions. 

Teaching Tips

Reinforcing the justification of non-differentiability is a key component of this lesson. Students have had much practice using limits to support continuity of the function f(x) and can easily understand that a discontinuity prevents the existence of a derivative. Justifying non-differentiability through the limits of f’(x) presents a notational challenge. Students will “see” a corner or a cusp and realize that a tangent line cannot exist at that point, but supporting that claim with limits of f’(x) will require much demonstration and practice.

Exam Insights

This is guaranteed AP Test content in both the multiple choice and free response sections. The time you take to lay the foundation (graphical, analytical, and verbal) for the concept of a derivative will be repaid in student retention, understanding and performance on the AP Test! As you review past AP questions, note the different representations, the different variables, and the varied notation presented to students. We can’t emphasize enough the importance of the experiences in 2.1 and 2.2.

Student Misconceptions

The limit expressions involving f(x) and f’(x) look the same to many students. Throw in “approaches from the left” or “approaches from the right” and some will be even more confused. Our language in the classroom must be precise and correct: we also have to work hard to avoid using “the function,” “the derivative,” or “it” during class discussion!

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