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 understand the concept of a derivative and evaluate derivatives of a function using the definition of derivative.

It is expected that students will:

• describe geometrically a secant line and a tangent line for the graph of a function at x = a.
• define and evaluate the derivative at x = a as: and
• define and calculate the derivative of a function using:
  
  
• use alternate notation interchangeably to express derivatives
• compute derivatives using the definition of derivative
• distinguish between continuity and differentiability of a function at a point
• determine when a function is non-differentiable, and explain why
• determine the slope of a tangent line to a curve at a given point
• determine the equation of the tangent line to a curve at a given point
• for a displacement function , calculate the average velocity over a given time interval and the instantaneous velocity at a given time
• distinguish between average and instantaneous rate of change

SUGGESTED INSTRUCTIONAL STRATEGIES

The concept of the derivative can be used to calculate instantaneous rate of change. Students can best understand this concept if it is presented to them geometrically, analytically, and numerically. This allows them to make connections between the various forms of presentation.

• Present an introduction to the "tangent line" problem by discussing the contribution made by Fermat, Descartes, Newton, and Leibniz.
• Use a graphing utility to graph
  On the same screen, add the graphs of , . Have students identify which of these lines appears to be tangent to the graph at the point (0, -5). Have them explain their answers.
• Sketch a simple parabola on the board. Attach a string at a point and show how it can be moved from a given secant line to a tangent line at a different point on the graph. Show how the slope of the secant line approaches that of the tangent line as .
• Have students calculate the slope of a linear function at a point , using the formula . Have students use the same formula to find the slope of the tangent lines to the graph of a non-linear function such as , and then find the equation of the tangent lines.
• Ask students to use technology to:
  • reinforce the result found when using the definition to calculate the derivative of a function
  • verify that the tangent line as calculated, appears to be the tangent to the curve. As an extension, have students zoom in at the point of tangency and describe the relationship between the tangent line and the curve
• Introduce the differentiability of a function by generating problems that require students to create, using technology, graphs with sharp turns and vertical tangent lines.
• Have students work in groups to prepare a presentation on tangent lines and the use of mathematics by the ancient Greeks (circle, ellipse, parabola). Ask them to include an explanation of how the initial work of the Greeks was surpassed by the work of Fermat and Descartes and how previously difficult arguments were made into essentially routine calculations.

SUGGESTED ASSESSMENT STRATEGIES

The concept of the derivative facilitates the solving of complex problems in fields such as science, engineering, and finance. Students should be able to demonstrate knowledge of the derivative in relation to problems involving instantaneous rate of change.

Observe

• As students work, circulate through the classroom and note:
  • the extent to which they relate their algebraic work to their graphing work
  • whether they can recognize the declining nature of the x values
• Have students develop a table of values to calculate a "slope predictor" in the parabola y=at a point. In small groups they can justify the concept of a limit using technology. Higher magnifications will eliminate the difference in the slope values of secant and tangent lines at a point. Ask one student from each group to explain this phenomenon, using the group’s ‘unique’ sample slope calculation. Note how succinctly they describe the processes used.
• When students work with technology (e.g., graphing calculator), check the extent to which they:
  • select appropriate viewing windows
  • enter the functions correctly
  • interpret the results correctly

Question

• Have students explain in their own words the concepts of average and instantaneous rates of change.

Research

• When students report on the contributions of various mathematicians to the tangent line problem, check the extent to which they:
  • clearly address key concepts (e.g., explain how Fermat found the length of the subtangent; how Descartes found the slope of the normal)
  • include careful graphic illustrations

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  pp. 83-85, 95-97, 105-107, 122-124, 152
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 60, 110, 112-117, 123, 125-129, 137, 142, 174, 662, 675
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 2 (Sections 2.7, 2.8, 2.9)

Multimedia

• Calculus: A New Horizon, Sixth Edition
  Ch. 3
  pp. 170, 172, 175, 178-180, 186, 187, 353
• Calculus of a Single Variable, Sixth Edition
  Ch. 2 (Section 2.1 )
  pp. 92-97, 108-109, 166

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

Ministry of Education Home Page