Lesson Plans

Calculus: Graphical, Numerical, Algebraic ©1999

by Finney

Week 22

Chapter 6, Section 6.1: Antiderivatives and Slope Fields; Section 6.2, Integration by Substitution

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.
• 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.
• Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, social, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the integral of a rate of change to give accumulated change or using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross-sections, the average value of a function, and the distance traveled by a particle along a line.
• Fundamental Theorem of Calculus.
  • Use of the Fundamental Theorem to evaluate definite integrals.
  • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
• Techniques of antidifferentiation.
  • Antiderivatives following directly from derivatives of basic functions.
  • Antiderivatives by substitution of variables (including change of limits for definite integrals).
• Applications of antidifferentiation.
  • Finding specific antiderivatives using initial conditions, including applications to motion along a line.
  • Solving separable differential equations and using them in modeling. In particular, studying the equation y ´ = ky and exponential growth.

Resources:

• Chapter 6, Section 6.1: Antiderivatives and Slope Fields; Section 6.2, Integration by Substitution—pp. 302–323
• Teacher's Guide with Answers—pp. 54–57
• Advanced Placement* Correlations and Placement–Concepts Worksheets—pp. 55–56
• Assessment—p. 57

Pacing Guide:

• 1st day—Section 6.1
• 2nd day—Section 6.2
• 3rd day—Section 6.2
• 4th day—Section 6.2
• 5th day—Section 6.2

Key Words:

• substitution method of integration
• separable differential equations

Designing Lessons/Student Responses to Lesson:

Critical Thinking Questions:

Troubleshooting Tips/Error Traps:

• Student errors in using the substitution method are legion. One of the most common is to insert the wrong constant multiplier. To prevent this type of mistake, emphasize the mechanical nature of the process, once the correct substitution is identified. For example, if u = 2x, then du = 2dx so we may solve for dx to obtain dx= du. Thus cos 2x • dx becomes cos u • du.
• Many different types of mistakes occur when substituting into definite integrals. Before starting a calculation, students should decide which of the two methods discussed in Exploration 2 is to be used and should use that method throughout. You may wish to encourage students to write the variable name with the limits of integration (e.g., u²du instead of u²du) in order to avoid mistakes caused by using the limits of x with the expression in u.
• When solving differential equations using separation of variables, students may not recognize alternate versions of s solution, particularly if the solution has been rewritten to isolate y.