Lesson Plans

Calculus: Graphical, Numerical, Algebraic ©1999

by Finney

Week 7

Chapter 3, Section 3.3: Rules for Differentiation; Section 3.4: Velocity and Other Rates of Change; Section 3.5: Derivatives of Trigonometric Functions

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.
• 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.
• Applications of derivatives.
  • Analysis of curves, including the notions of monotonicity and concavity.
  • Optimization, both absolute (global) and relative (local) extrema.
  • Modeling rates of change, including related rates problems.
  • Use of implicit differentiation to find the derivative of an inverse function.
  • Interpretation of derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
• 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 3, Section 3.3: Rule for Differentiation; Section 3.4: Velocity and Other Rates of Change; Section 3.5: Derivatives of Trigonometric Functions—pp. 112–141
• Teacher's Guide with Answers—pp. 23–28
• Advanced Placement* Correlations and Placement–Concepts Worksheets—pp. 30–33
• Assessment—none

Pacing Guide:

• 1st day—Section 3.3
• 2nd day—Section 3.4
• 3rd day—Section 3.4
• 4th day—Section 3.4
• 5th day—Section 3.5

Key Words:

Designing Lessons/Student Responses to Lesson:

• 3.4—You can open this lesson by recalling Example 1 of Section 2.1, in which the instantaneous speed of a falling rock was determined. Explain the difference between speed and velocity, and present velocity as an example of a rate of change. The main topic of this section is the motion of a particle along a straight line, with emphasis on free-fall motion. The concepts of velocity, speed, and acceleration are essential to an understanding of this section. Emphasize that velocity is the rate of change of position, and acceleration is the rate of change of velocity. The acceleration due to gravity can be either a positive or a negative number depending on how one defines a coordinate system. The discussion of marginal cost and marginal revenue is an excellent way to conclude the lessons. Students should recognize that many rates of change do not involve time nor position.
• 3.5—You can begin this lesson by having students complete Exploration 1, which allows students to guess rules for the derivatives of the sine and cosine functions. In connection with this exploration, you may wish to have students set h = 0.001 and graph the difference quotient, . Students should recognize the graph of y = cos x. The rule for differentiating sin x is proved directly from the definition of the derivative, using the two fundamental limits = 1 and = 0. The rule for differentiating cos x can be derived in a similar manner. You may wish to point out the important role of the angle sum formulas for sin(a + b) and cos(a + b) in these derivatives. The rules for differentiating the other four basic trigonometric functions can be shown easily by using the identities and the rules for differentiating the sine and cosine functions.

Critical Thinking Questions:

• 3.4—pp. 130–133, # 16–17, 23–25, 28–29
  Have students graph y = 0.5x³ – 3x in a [–4, 4] by [–3, 3] window. Ask: If f ´(x) = 0.5x³ – 3x , what does the graph tell you about the function of f (x)?
• 3.5—p. 140, # 28, 35

Troubleshooting Tips/Error Traps:

• Some students have trouble distinguishing between speed and velocity.
• Students sometimes forget to misapply the basic trigonometric identities. You may wish to review the reciprocal, Pythagorean, angle sum, and half-angle identities.
• When using a grapher, students sometimes forget to use radian mode or to enter expressions such as cos² x as (cos x)².