Calculus: Graphical, Numerical, Algebraic ©1999
Chapter 2, Section 2.3: Continuity; Review Exercises: Limits and Continuity
College Board Objectives:
- Analysis of graphs.
- Limits of functions (including one-sided limits). An intuitive understanding of the limiting process is sufficient for this course.
- Calculating limits using algebra.
- Estimating limits from graphs or tables of data.
- Continuity as a property of functions. The central idea of continuity is that close values of the domain lead to close values of the range.
- Understanding continuity in terms of limits.
- Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
- Derivative at a point.
- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
- Tangent line to a curve at a point and local linear approximation.
- Instantaneous rate of change as the limit of average rate of change.
- Approximate rate of change from graphs and tables of values.
- Chapter 2, Section 2.3—Review Exercises: Continuity , pp. 73–93
- Teacher's Guide with Answers—pp. 15–18
- Advanced Placement* Correlations and Placement–Concepts Worksheets—pp. 26–27
- Advanced Placement* Correlations and Placement–Group Activity Exploration—pp. 78–80
- Assessment—p. 8
- 1st day—Section 2.3
- 2nd day—Section 2.3
- 3rd day—Section 2.4
- 4th day—Section 2.4
- 5th day—Review Exercises
- continuous on an interval
- removable discontinuity
- jump discontinuity
- oscillating discontinuity
- Intermediate Value Theorem
- secant line
- secant line slope
- tangent to a curve
- difference quotient
- instantaneous rate of change
- continuous as a point
- non-removable discontinuity
- infinite discontinuity
- continuous function
- average rate of change
- tangent line
- tangent line slope
- slope of a curve
- instantaneous speed
Designing Lessons/Student Responses to Lesson:
- 2.3—You can begin the lesson by using a pencil to trace the graph of a continuous function over any interval without lifting the pencil from the paper. The definition of continuity must be understood by students if they are to be successful in this course. Be careful to distinguish between continuity at an interior point in the domain (which involves a two-sided limit) and continuity at an endpoint (which involves a one-sided limit). Different types of discontinuities are discussed on page 76. It will be worthwhile to illustrate each type of discontinuity with a grapher or with a chalkboard sketch. Names for the different types of discontinuities should be used consistently throughout the course.
- 2.4—Most students should already have an understanding of slopes, so they should be encouraged to examine the graphs of functions to make sure that their answers are reasonable. The average rate of change introduced in this lesson provide an application related to limits or rational functions.
Critical Thinking Questions:
- 2.3—p. 81, # 45
- 2.4—pp. 89–90, # 35–43
Troubleshooting Tips/Error Traps:
- Some students may have difficulty recognizing the differences between continuity at an interior point in the domain and continuity at an endpoint.
- Some students may mistakenly attempt to apply the Intermediate Value Theorem to functions that are not continuous.
- Many students are prone to algebraic errors when calculating the slopes of curves. Encourage students to understand their own mistakes so that they will be less likely to make these mistakes in the future.
End of Chapter Activity—enrichment for high-level critical thinking:
Chapter 2, Review Exercises, pp. 91–93, # 25–26, 49, 53