Lesson Plans

Calculus: Graphical, Numerical, Algebraic ©1999

by Finney

Week 3

Chapter 1, Review Exercises: Prerequisites for Calculus
Chapter 2, Section 2.2: Limits Involving Infinity

College Board Objectives:

• Analysis of graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
• 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.
• Asymptotic and unbounded behavior.
  • Understanding asymptotes in terms of graphical behavior.
  • Describing asymptotic behavior in terms of limits involving infinity.
  • Comparing relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.)
• 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.

Resources:

• Chapter 1, Review Exercises: Prerequisites for Calculus
• Chapter 2, Section 2.2: Limits Involving Infinity—pp. 52–73
• Teacher's Guide with Answers—pp. 11–15
• Advanced Placement* Correlations and Placement–Concepts Worksheets—pp. 26–27
• Assessment—pp. 3–7

Pacing Guide:

• 1st day—Test
• 2nd day—Section 2.1
• 3rd day—Section 2.1
• 4th day—Section 2.2
• 5th day—Section 2.2

Key Words:

Designing Lessons/Student Responses to Lesson:

• 2.1—The function y = can provide an excellent introduction to the idea of limits. Using a grapher, students can see that this function approaches 1 as x approaches 0. This discussion is found on page 56. An important goal of the course is to make students be at ease with technology for exploration, confirmation and interpretation of results, and problem solving. Graphing utilities are very useful in the study of functions, but they cannot be used for proofs. Stress the importance of confirming all graphical solutions by algebraic methods. A discussion of Exercise 54 and the Sandwich Theorem is a good way to conclude this lesson. Students should understand the Sandwich Theorem intuitively.
• 2.2—A discussion of the function y = provides a meaningful introduction to both limits as x and infinite limits as x approaches a constant. The mathematical meaning of the symbol should be understood in the context of the phrase "x ," meaning that in a function of x, the x-value increases without bound. Students should understand that does not represent a real number. You may wish to discuss the end behavior of three types of rational functions; that is, rational functions where the degree of the numerator is less than, equal to, and greater than the degree of the denominator, respectively.

Critical Thinking Questions:

• 2.1—pp. 64–65, # 59–66
  Have students write a paragraph describing any difficulties they encountered while finding limits.
• 2.2—pp. 72–73, # 49–50, 53–60
  Ask students to express their understanding of end behavior models in their own words.

Troubleshooting Tips/Error Traps:

• Graphing utilities sometimes connect the two branches of the graph of a function like f(x) = , suggesting that the function is defined and continuous for every value in the domain, including x = –1. To avoid this spice or phantom asymptote in the display, you can use a "friendly window" or "decimal window" (that is, a window in which the pixels correspond to exactly 0, 0.1, 0.2, and so on), or use the dot mode format for the display of graphs of such functions.
• Many students will have trouble with limits involving the indeterminate forms , , and 0 • . L'Hopital's rule, a rigorous approach to evaluating these limits, is the subject of Section 8.1.