Supporting a Coherent Understanding of Limits, Derivatives, and Integrals through Approximation (CoRD)
Michael Oehrtman, Oklahoma State University
Michael Tallman, Oklahoma State University
These resources are designed to be used through a first-semester calculus course to help students develop a rigorous and productive understanding of limits, derivatives, and integrals. The activities engage students in problem-solving about approximating unknown quantities using appropriate function values, difference quotients, or Reimann sums. While these ideas reflect the underlying structure of formal ε-δ definitions and proofs, they are framed in more accessible language and ideas of errors, error bounds, and reasoning to achieve any desired degree of accuracy. Representing their reasoning in consistent ways across multiple representations helps students link these situation-specific ideas to the underlying mathematical principals in a coherent way. Then engaging in similar tasks across different contexts supports attention to the general mathematical structure of limits, derivatives, and definite integrals that can support subsequent modeling and interpretation.
These resources were adapted from CLEAR Calculus and The Calculus Videos Project under the creative commons Attribution-NonCommercial-ShareAlike copyright CC BY-NC-SA 4.0
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Constant and Average Rate of Change: A Rocket Launch
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These slides are intended to lead a whole-class discussion reviewing the concepts of constant and average rate of change, while setting up the problem of an instantaneous rate.
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Limit of a Function at a Point: Locate the Hole |
In this activity, students explore functions with removable discontinuities to develop a graphical, numerical and algebraic sense of the meaning of the limit of a function at a point. By approximating the “height of the hole” to an arbitrary degree of accuracy, they develop the terminology and structure of errors and error bounds that will serve as the overarching structure for everything defined in terms of a limit in the course.
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Video Discussion of the Limit of a Function at a Point: Locate the Hole |
In this video, we introduce the concept of limit at a point by considering the limiting value of a function as the independent variable approaches the abscissa of a removable discontinuity. We support students in (1) identifying the quantity we are trying to approximate (i.e., height of a “hole” on a graph in Cartesian coordinates), (2) generating over- and under-estimates of this quantity, (3) computing an error bound as the difference of over- and under-estimates, and (4) recognizing that this error bound can be reduced as much as desired by making the controlling variable closer to the singularity.
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The Derivative of a Function at a Point, Part 1: The Instantaneous Speed of a Crossbow Bolt |
This task is intended to introduce students to the concept of instantaneous rate of change and the derivative. By attempting to approximate the speed of a crossbow bolt fired into the air at a given moment, students will encounter using the average rate of change over a small time interval as an approximation. Attending to the gravitational force slowing the bolt down students may find both underestimates and overestimates, and thus bound the error for their approximations. Achieving any desired degree of accuracy supports reasoning about rich the limit structure in a natural way as a response to the problem. Students explore and explain their reasoning in multiple representations the reinforce a coherent understanding of the derivative concept.
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The Derivative of a Function at a Point, Part 2: Radioactive Decay in Medical Iodine-123 |
In this task, students repeat the same mathematical problem-solving as in Part 1, but in a new context, the decay of Iodine-123 in a diagnostic medical treatment. Recognition of the common structure of their activity across the two tasks is intended to support students in moving beyond thinking of the derivative as modeling only speed or slope of a tangent to reasoning about instantaneous rate of change more generally.
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Video Discussion of Instantaneous Rate of Change as the Limit of Average Rates |
In this video, we support students’ understanding of instantaneous rate of change as the limiting value of average rates of change as the interval of input values over which these average rates of change approach (but do not become) zero. We prompt students to consider how to quantify the instantaneous speed of an object, and support them in specifying the criteria by which they can conclude that their approximations are over- or under-estimates of the instantaneous speed of the object. As with the video about the limit of a function at a point, we support students’ understanding that we can make the error bound of our approximations as small as desired by making the controlling variable closer to the singularity.
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Definite Integrals, Part 1: Distance Traveled by a Lunar Rover |
This task is intended to introduce students to the concept of accumulation and the definite integral. By attempting to approximate the distance traveled by a lunar rover moving with a varying speed, students will encounter using Riemann sum with a fine partition as an approximation. Attending to the acceleration, students may find both underestimates and overestimates, and thus bound the error for their approximations. Achieving any desired degree of accuracy supports reasoning about rich the limit structure in a natural way as a response to the problem. Students explore and explain their reasoning in multiple representations the reinforce a coherent understanding of the definite integral concept.
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Video Discussion of Accumulation of Dust on a Mars Rover (Suggestions for Incorporating Videos into Instruction)
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This video explains what a definite integral is in terms of total accumulation of a quantity, shows how to write definite integrals, and explains how they’re related to Riemann sums.
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Definite Integrals, Part 2: Mass in an Oil Spill |
In this task, students repeat the same mathematical problem-solving as in Part 1, but in a new context, the mass of an oil spill in the ocean with varying density as a function of the distance from the spill. Recognition of the common structure of their activity across the two tasks is intended to support students in moving beyond thinking of the integral as modeling only distance traveled or signed area “under a graph” to reasoning about accumulation more generally.
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Video Discussion of Accumulation of Mass in an Oil Spill (Suggestions for Incorporating Videos into Instruction)
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This video provides another example of writing a definite integral to represent the amount of a quantity that has accumulated, starting with a Riemann sum.
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This work is licensed under CC BY-NC-SA 4.0