Use formal limit definitions of continuity and differentiability
I can find values that make a function continuous and differentiable
Quick Lesson Plan
Today students are working in pairs on another excellent activity from Bryan Passwater. Partner A and Partner B work on different problems of the same type but whose answers MATCH, allowing for students to self-check the reasonableness of their solutions and have valuable discussions.
Split your class into pairs and have students decide who is Partner A and who is Partner B. Partner A will get one set of problems while Partner B gets the other. Students should work on their individual problem and then check with their partner to confer. If answers do not match, have partners work together to find the error and correct their answers.
You may wish to use the final 15 minutes of class to review the definition of the derivative since the Twinning activity focuses exclusively on Topics 2.3-2.4. An important skill for students is recognizing the limit definition and evaluating the limit using derivatives instead of other analytical methods. You can scaffold this process by having students practice identifying the function in question, whether the limit is asking for a general derivative equation or the derivative at a point, and then evaluating the derivative. We have students practice looking at a limit definition and then fill in the sentence “[limit expression] is asking for…”
Continuity and differentiability is an important idea tested on the AP Exam! Students must be able to justify continuity and differentiability using limit definitions, identify places of discontinuity and non-differentiability from a graph, and find values that would make a function continuous or differentiable at a given x-value. Free response questions require students to understand how continuity and differentiability are tied to the hypotheses of theorems such as the IVT and MVT.
The trickiest problems for our students was question 4 on Partner A’s sheet and question 14 on Partner B’s sheet. Students could not get their answers to match! The correct answer to both questions is “Not Possible”. What students failed to notice is the lack of an equality symbol in Partner B's domain, meaning that the function was not defined at the given x-value, which contradicts definitions for continuity and thus also differentiability.