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 antidifferentiation to solve a variety of problems.

It is expected that students will:

• use antidifferentiation to solve problems about motion of a particle along a line that involve:
  • computing the displacement given initial position and velocity as a function of time
  • computing velocity and/or displacement given suitable initial conditions and acceleration as a function of time
• use antidifferentiation to find the area under the curve , above the x-axis, from x = a to x = b.
• use differentiation to determine whether a given function or family of functions is a solution of a given differential equation
• use correct notation and form when writing the general and particular solution for differential equations.
• model and solve exponential growth and decay problems using a differential equation of the form:
• model and solve problems involving Newton’s Law of Cooling using a differential equation of the form:

SUGGESTED INSTRUCTIONAL STRATEGIES

Antidifferentiation enables students to solve problems that are otherwise unsolvable (e.g., problems related to exponential growth and decay). These problems are found in a variety of contexts, from science to business.

• Demonstrate how physics formulae about motion under constant acceleration can be justified using antidifferentiation. If for some C, and therefore for some D. The two constants of integration can be found by using initial conditions. Given g = 9.81 and a rock thrown upward at a given speed from a 100 m tower, it is possible to calculate the maximum height reached and the time it takes for the rock to hit the ground. There are many variations on this problem.
• To explain why antiderivatives can be used to solve area problems, let f(x) be some given function, and let A(u) be the area under , above the x axis, from . By looking at , we can argue reasonably that , at the same time reinforcing the concept of derivative from the definition. So is an antiderivative of
• Many calculators have a numerical integration feature that approximates . Have students find the area from a to b under a curve for which they can find . Compare this solution with the calculator’s approximation.
• Give students a function such as and ask them to make a rough sketch of an antiderivative .
• Illustrate the wide applicability of the concepts by using examples that are not from the physical sciences. For example, let be the cost of producing x tons of a certain fertilizer. Suppose that . The cost of producing 2 tons is $5000. What is the cost of producing 100 tons?
• Have students generate and maintain a list of phenomena other than radioactive decay that are described by the same differential equation. For example, the illumination that reaches x metres below the surface of the water can be described by . Under constant inflation, the buying power of a dollar t years of now can be described by .

SUGGESTED ASSESSMENT STRATEGIES

An understanding of antidifferentiation and its applications is essential to the solution of the "area under the curve" problems. Students can demonstrate their understanding of antidifferentiation ideas and skills through their problem-solving work.

• To check whether the meaning of the phrase "solution of a differential equation" is fully understood, have students work in groups to discuss and solve problems such as the following:
  • show that is a solution of the differential equation
  • find a solution of the above differential equation with
• When students are asked to solve problems such as the following, verify that they are able to identify sin kt and cos kt as solutions of the differential equation and that they can find other solutions as well:
  • A weight hangs from an ideal spring. It is pulled down a few centimetres then released. If y is the displacement of the weight from its rest position, it turns out that for some constant k
• Take a problem (e.g., an exponential decay problem with several parts) and ask for the solution to be written up as a prose report, in complete sentences, with the reasons for each step clearly explained. Criteria for assessment include clarity and grammatical correctness.
• Pose the problem, "In how many different ways can we find the area under , above the x axis, from , exactly or approximately?" Students can write a report on this. Students should be able to identify
  • Archimedes’ method
  • standard antiderivative method
  • approximation techniques of their own devising
• Ask students to use standard Internet resources to find information about differential equations in various areas of application, and to report on an area of interest to them. Assess the extent to which they are able to express their findings coherently and in their own words.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic
  Ch. 6 (Section 6.1, 6.4)
  pp. 262 - 263, 311, 333-334
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 278, 290, 373-374, 377, 381-382, 385, 409
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 4 (Section 4.10)
  Ch. 5 (Sections 5.1, 5.4)
  pp. 586, 603, 611

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 328, 408, 433, 580, 601, 611
• Calculus of a Single Variable, Sixth Edition
  Ch. 4 (Section 4.4)
  Ch. 5 (Section 5.5)
  pp. 242, 362

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

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