Removing Discontinuities (Topic 1.13)
Unit 1 - Day 10
Unit 1
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Day 11
Day 12
Day 13
Day 14
Day 15
Day 16
Day 17
Day 18
All Units
Learning Objectives
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Determine locations of removable discontinuities by graphical, numeric, or analytic methods
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Determine when and how discontinuous functions can be made continuous
Success Criteria
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I can locate removable discontinuities by using the definitions of limits and continuity.
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I can calculate the needed function value to retain a limit and create continuity.
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I can use extended functions to define or redefine the y-value at a point to match the limit at that point.
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I can use the definition of continuity to justify my solutions.
Quick Lesson Plan
Activity: Can This Date Be Salvaged?
Lesson Handout
Answer Key
Overview
This lesson follows the adventures of Pia and Pepe at Starbucks. Previously, we examined types of discontinuities. Today, students define points, if possible, to remove a discontinuity where a limit exists.
Teaching Tips
Assign work with piecewise functions often to highlight domain restrictions, graphing techniques, and behavior of the function at important domain values. Continue to require demonstration of the definition of continuity, especially when using analytic methods.
Exam Insights
Remind students that the AP Exam will ask them to evaluate function behavior at “interesting” values of the domain. In the practice exam question here, x = 2 is the chosen value, of course!
Student Misconceptions
After following rigid mathematical rules for years, many of us (teachers included!) are troubled by the idea that we can reassign given values of a function or simply calculate a function value to create continuity. When this is requested, however, encourage students to check their calculations with both definitions to confirm that the value they have found equals the limit from each side.