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 represent and analyze rational, inverse trigonometric, base e exponential, natural logarithmic, elementary implicit, and composite functions, using technology as appropriate.

It is expected that students will:

• model and apply inverse trigonometric, base e exponential, natural logarithmic, elementary implicit and composite functions to solve problems
• draw (using technology), sketch and analyze the graphs of rational, inverse trigonometric, base e exponential, natural logarithmic, elementary implicit, and composite functions for:
  • domain and range
  • intercepts
• recognize the relationship between a base a exponential function (a > 0) and the equivalent base e exponential function
• determine, using the appropriate method (analytic or graphing utility) the points where

SUGGESTED INSTRUCTIONAL STRATEGIES

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.

• Prior to undertaking work on functions and their graphs, conduct review activities to ensure that students can:
  • perform algebraic operations on functions and compute composite functions
  • use function and inverse function notation appropriately
  • determine the inverse of a function and whether it exists
  • describe the relationship between the domain and range of a function and its inverse
• Give students an exponential function and its inverse logarithmic function. Have them work in groups, using graphing calculators or computer software, to identify the relationship between the two. (e.g., compare and contrast the graphs of ).
• Emphasize the domain and range restrictions as students work with exponential and natural logarithmic functions.
• Have students use technology (e.g., graphing calculators) to compare the graphs of functions such as:
  
  
  
• Similarly apply this procedure to trigonometric and inverse trigonometric graphs including: sine and arcsine, cosine and arccosine, and tangent and arctangent. Emphasize that arcsine and does not mean the reciprocal of sin x

SUGGESTED ASSESSMENT STRATEGIES

The exponential and natural logarithmic functions enable students to solve more complex problems in areas such as science, engineering, and finance. Students should be able to demonstrate their knowledge of the relationship between natural logarithms and exponential functions, both in theory and in application, in a variety of problem situations.

Observe

• Ask students to outline how they would teach a classmate to explain:
  • the inverse relationship between base e exponential and natural logarithmic functions
  • the restrictions associated with base e exponential and natural logarithmic, and trigonometric and inverse trigonometric functions
    Note the extent to which the outlines:
  • include general steps to follow
  • use mathematical terms correctly
  • provide clear examples
  • describe common errors and how they can be avoided

Collect

• Assign a series of problems that require students to apply their knowledge of the relationship between logarithms and exponential functions. Check their work for evidence that they:
  • clearly understood the requirements of the problem
  • used efficient strategies and procedures to solve the problem
  • recognized when a strategy or procedure was not appropriate
  • verified that their solutions were accurate and reasonable

Self-Assessment

• Discuss with students the criteria for assessing graphing skills. Show them how to develop a rating scale. Have them use the scale to assess their own graphing skills.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  pp. 12, 16, 23, 36-37, 45-51, 165
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  Ch. 1 (Sections 1.4, 1.5, 1.6)
  Ch. 4 (Section 4.6)
  pp. 5, 20, 26-27, 41, 51-52, 63, 158, 182
• Single Variable Calculus Early Transcendentals, Fourth Edition
  pp. 43-46, 61-63, 69-72, A31-A34

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 49, 57, 240, 242-243, 246, 254, 261, 308
• Calculus of a Single Variable, Sixth Edition
  Ch. 5 (Section 5.1, 5.4, 5.5, 5.8)
  pp. 20, 26, 27, 75, 134, 380

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 5, 2000

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