Lesson Plans

Calculus: Graphical, Numerical, Algebraic ©1999

by Finney

Week 5

Chapter 2: Test: Limits and Continuity
Chapter 3: Section 3.1: Derivative of a Function; Section 3.2: Differentiability

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.
• Concept of the derivative. The concept of the derivative is presented geometrically, numerically, and analytically, and is interpreted as an instantaneous rate of change.
  • Derivative defined as the limit of the difference quotient.
  • Relationship between differentiability and continuity.
• 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.
• Derivative as a function.
  • Corresponding characteristics of graphs of f and f ´.
  • Relationship between the increasing and decreasing behavior of f and the sign of f ´.
  • The Mean Value Theorem and its geometric consequences.
  • Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
• Computation of derivatives.
  • Knowledge of derivatives of basic functions, including x^r, exponential, trigonometric, and inverse trigonometric functions.
  • Basic rules for the derivative of sums, products, and quotients of functions.
  • Chain Rule and implicit differentiation.

Resources:

• Chapter 2 Test: Limits and Continuity;
• Chapter 3, Section 3.1: Derivative of a Function; Section 3.2: Differentiability—pp. 94–112
• Teacher's Guide with Answers—pp. 19–23
• Advanced Placement* Correlations and Placement–Concepts Worksheets—none
• Advanced Placement* Correlations and Placement–Group Activity Exploration–p. 81
• Assessment—pp. 9–12

Pacing Guide:

• 1st day—Test
• 2nd day—Section 3.1
• 3rd day—Section 3.1
• 4th day—Section 3.1
• 5th day—Section 3.2

Key Words:

Designing Lessons/Student Responses to Lesson:

• 3.1—An interesting way to begin this lesson is by presenting the graph of a function and showing how the slope of secant lines approaches a limit corresponding to the slope of a tangent line. Students should learn to calculate derivatives using the definition. This can be done in two stages: first, calculate the derivative at a particular point x = a, then generalize the process to calculate the derivative at a generic point x. The different notations for derivatives should be discussed. The lesson can be concluded by discussing one-sided derivatives. When discussing one-sided derivative of a piecewise defined function, emphasize that you are evaluating the left- and right-hand derivatives of a single function (not two functions).
• 3.2—You can introduce this lesson with an informal discussion of what it means for a function to be differentiable or nondifferentiable at a point. Give several examples to illustrate corners, cusps, vertical tangents, and discontinuities. The text denotes a numerical approximation to f ´(a) by the symbol NDER f(a) or NDER (f(x), a). Each student should be able to use his grapher to calculate numerical derivatives. A discussion on the Intermediate Value Theorem for derivatives can be an effective way to conclude the lesson.

Critical Thinking Questions:

• 3.1—p. 104, # 26–29
  Have students use the technique in Exercise 29 to create tables showing the probability that, in a group of n people, at least two people will have birthdays in the same month. Explain what happens when n 13.
• 3.2—p. 112, # 23, 31
  In Exercise 2, the points (0, 2) and (2, 4) are on the graph and 1 is between f ´(0) and f ´(2), but there is no value of x where f ´(x) = 1. Have students explain why this does not violate the Intermediate Value Theorem for Derivatives.

Troubleshooting Tips/Error Traps:

• When calculating derivatives using the definition, students often make errors in evaluating and simplifying the numerator of the difference quotient. When f(x) is a polynomial or rational function, h is always a factor of the simplified expression
• Students must know when f ´(a) fails to exist.