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Lesson Plans
Calculus: Graphical, Numerical, Algebraic ©1999
by Finney
Week 8
Chapter 3, Section 3.5: Derivatives of Trigonometric Functions; Section 3.6: Chain Rule
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.
- 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 xr, 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 3.5: Derivatives of Trigonometric Functions; Section 3.6: Chain Rule—pp. 134–149
- Teacher's Guide with Answers—pp. 26–29
- Advanced Placement* Correlations and Placement–Concepts Worksheets—none
- Assessment—p. 14
Pacing Guide:
- 1st day—Section 3.5
- 2nd day—Section 3.5
- 3rd day—Section 3.6
- 4th day—Section 3.6
- 5th day—Section 3.6
Key Words:
- simple harmonic motion
- Chain Rule
- jerk
- power chain rule
Designing Lessons/Student Responses to Lesson:
3.6—The text takes a traditional approach for teaching the correct usage of the Chain Rule. First, students are taught to differentiate y = f (g(x)) by setting u = g(x), calculating the two derivatives f ´(u) and g ´(x), and then applying the Chain Rule to obtain y ´ = f ´(u) • g ´(x) = f ´(g(x)) • g ´. The process is then shortened by dispensing with the u and simply referring to g(x) as the inside function. This abbreviated process is called the outside-inside rule. Students should get plenty of practice with the Chain Rule so that its use becomes automatic. When presenting the Leibniz form of the Chain Rule, 
, emphasize that 
is evaluated at u = g(x) and 
is evaluated at x.
Critical Thinking Questions:
3.6—p. 149, # 69–71
Troubleshooting Tips/Error Traps:
In applying the outside-inside rule to differentiate f(g(x)), a common mistake is to omit the factor g ´(x) in the answer.