Unit 9: Limits and Continuity
Students are introduced to calculus topics of limits, continuity, and derivatives, and gain a foundation of essential calculus skills.
Unit 9, Limits and Continuity, is an introduction to the calculus topics of limits, continuity, and derivatives, and it provides a foundation in the essential calculus skill of thinking and reasoning about the infinitely small and the infinitely large while also arguing logically based on definitions and theorems. Students should leave this unit confident working with piecewise functions, finding finite and infinite limits of various types of functions graphically and algebraically, and rigorously defining continuity.
This unit also offers a useful opportunity to review and deepen knowledge of various types of functions, and rational functions in particular offer a useful opportunity to apply knowledge of limits, analyze asymptotes and removable discontinuities, and think about end behavior, while all function types help students to apply function transformations in different contexts. These skills are essential foundational knowledge for calculus, and this unit should be tailored to the specific strengths and weaknesses of students to maximize the utility of that flexibility.
If there is not time for a nine-lesson unit at this point in the year, Lessons 3 and 5 could be cut. If there is extra time, Lesson 5 could be expanded to practice more parent functions, and Lessons 7 and 8 could be expanded to practice working with rational functions, factoring, and manipulating functions algebraically. If the unit is shortened, consider cutting corresponding questions from the assessment as well.
Since this unit serves as an introduction to Calculus, the lessons are aligned to the Learning Objectives and Essential Knowledge described in the College Board's AP Calculus AB and AP Calculus BC Course and Exam Description in the place of Common Core State Standards.
Pacing: 11 instructional days (9 lessons, 1 flex day, 1 assessment day)
The following assessments accompany Unit 9.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Suggestions for how to prepare to teach this unit
Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
The central mathematical concepts that students will come to understand in this unit
The materials, representations, and tools teachers and students will need for this unit
Desmos: This unit requires generating numerous piecewise graphs with specific features, and Desmos is the easiest tool to create these graphs.
Terms and notation that students learn or use in the unit
Topic A: Limits and Continuity
Graph, write, and evaluate linear piecewise functions.
Use interval and function notation to describe the behavior of piecewise functions.
Sketch a slope graph from a linear piecewise function.
Find limits, including left- and right-hand limits, on a function given graphically.
Define continuity of functions and determine whether a function is continuous on a particular domain.
Write and evaluate piecewise functions algebraically and graphically using parent functions.
State and evaluate limits algebraically.
Evaluate infinite limits and limits at infinity.
Sketch functions given limits and continuity requirements.
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— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Probability and Statistical Inference