This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will use the first and second derivatives to describe the characteristic of the graph of a function.

It is expected that students will:

• given the graph of :
  • 
  • relate the sign of the derivative on an interval to whether the function is increasing or decreasing over that interval.
  • relate the sign of the second derivative to the concavity of a function.
• determine the critical numbers and inflection points of a function.
• Determine the maximum and minimum values of a function and use the first and/or second derivative test(s) to justify their solutions
• use Newton’s iterative formula (with technology) to find the solution of given equations,
• use the tangent line approximation to estimate values of a function near a point and analyze the approximation using the second derivative.

SUGGESTED INSTRUCTIONAL STRATEGIES

The first and second derivatives of a function provide a great deal of information concerning the graph of the function. This information is critical to solving problems involving calculus and helps students understand what the graph of a function represents.

• Ask students to compare the questions, "Where is - increasing most rapidly? Where is - increasing?" Have students develop similar questions.
• Have students work in teams of two to determine why the calculator is not helpful in questions such as: " . Describe in terms of the parameter a, where reaches a maximum or minimum."
• Have students use technology to explore features of functions such as the following:
  • when a function has maximum/minimum points or neither
  • where the inflection points occur
  • where curves are concave up or down
  • vertical or horizontal tangent lines
  • the relationship among graphs of the 1st, 2nd derivatives of a function and the function
  • the impact that endpoints have on the maximum/minimum
• Use a summary chart to demonstrate increasing and decreasing aspects of a function. Have students describe how this relates to the 1st derivative.
• Have groups of students learn particular concepts, (e.g., Newton’s method) and teach these to their peers. Have them examine situations where it works very efficiently, situations where it works very slowly , and situations where it does not work . Challenge them to develop hypotheses as to why it does or does not work.
• Have students research and report on the history of Newton’s method for determining square roots (e.g., mentioning Heron of Alexandria, and the work of mathematicians in Babylon, India).
• Discuss with students the relative merits of using or not using a graphing calculator "to do" calculus. Present them with questions such as contrasted with . Note that the calculator does not always give the necessary detail.

SUGGESTED ASSESSMENT STRATEGIES

Students demonstrate their understanding of first and second derivatives by relating the information they obtain from these to the graph of a function (i.e., critical numbers, inflection points, maximum and minimum values, and concavity).

Observe

• When reviewing students work, note the extent to which they can:
  • accurately determine the 1st and 2nd derivative
  • apply the respective tests
• recognize and describe the relationship among the graphs of

Collect

• Ask the students to sketch a cubic graph and determine what the slopes of the tangent line would be at the local maximum and minimum points. Have them present their summary to the class. Consider the extent to which they can explain why the derivative is 0 at a local maximum/minimum.

Peer Assessment

• Have students critique each others’ graphs using criteria generated by the class. These criteria could include the extent to which:
  • a graph is appropriate to the function it is supposed to represent
  • the axes are accurately labelled
  • appropriate scales have been chosen for the axes
  • the graphs present smooth curves
  • domain, range, asymptotes, intercepts, and vertices have been correctly determined
  • inflection points, maximum and minimum points, and region of concavity have been correctly identified

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  Ch. 4 (Sections 4.4, 4.6)
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 82, 127-128, 173, 247
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 3 (Section 3.10)
  Ch. 4 (Sections 41, 4.7)

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 172, 270, 329
  Calculus of a Single Variable, Sixth Edition
  Ch. 2 (Section 2.6)
  Ch. 3 (Section 3.7)

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 7, 2000

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