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Visualizing Related Rates

Author:Annauen Joy Abellana Empaces

GeoGebra Applet Introduction: Exploring Related Rates and Implicit Differentiation

This GeoGebra applet is designed to visualize and explore the concept of related rates and implicit differentiation. By manipulating the slider variable t we analyze the relationship between points A=(2t,0) and B=(0,7t) and observe how changes in x and y are connected as t varies. Key Elements of the Applet
  1. Points and Vectors:
    • A=(2t,0): Represents the change in x with respect to t.
    • B=(0,7t): Represents the change in y with respect to t.
    • Vector u=Vector(C,B) and v=Vector(C,A); Show the directional changes in y and x from the origin C=(0,0)
    • Segment: Represents the distance between points A and B, visually connecting the changes in x and y. Visualize points A and B as cars driving at different speed.
  2. Related Rates:
    • By exploring the rate dx/dy=2/7 or dy/dx​=7/2​, we study how a small change in x corresponds to changes in y. This reflects the concept of related rates, where each change in x (e.g., Δx=0.0001) is accompanied by a proportional change in y (e.g., Δy=0.00035) reinforcing that the rate dx/dy is approximately 0.29.
  3. Implicit Differentiation and Slope:
    • The applet also highlights implicit differentiation by examining the slope dy/dx=3.5, meaning y changes 3.5 times faster than x for each unit change. This reciprocal relationship is derived from x=2/7y, providing insight into the interconnected rates and slope.
Purpose and Learning Outcomes This interactive applet enables users to:
  • Visualize how changes in one variable affect another through related rates.
  • Observe the behavior of Δx\Delta and Δy\Delta y as t varies and understand the calculated slope dy/dx=3.5
  • Gain an intuitive understanding of the mathematical principles of related rates and implicit differentiation in a dynamic, hands-on environment.
With this applet, we aim to deepen comprehension of how variables interrelate through rates of change, demonstrating fundamental calculus concepts in a clear and engaging way. Acknowledgment: Special thanks to ChatGPT for enhancing the articulation of this introduction based on the discovered GeoGebra applet design.

Source of Inspiration

How to Use This GeoGebra Applet

  1. Move the Slider t:
    • Adjust the slider t to see how points A=(2t,0) and B=(0,7t) move as t changes.
    • Observe how the corresponding changes in x and y align with the given rates dy/dx=3.5 and dx/dy=0.2857 (the inverse of 3.5).
  2. Validate Implicit Differentiation:
    • Note the slope dy/dx=3.5, representing how y changes 3.5 times faster than x.
    • Practice calculating dx/dy ​ by observing that x changes at a rate of 0.28 times for each unit change in y, reinforcing the relationship.
Experiment with the slider to explore how the changes in x and y validate the implicit differentiation results, gaining insight into related rates and slope.

Slider t

What does moving the slider t in this applet primarily demonstrate?

Select all that apply
  • A
  • B
  • C
  • C) The concept of area under a curve

  • D
    • Check my answer (3)

    Slope dy/dx=3.5

    What does the slope dy/dx=3.5 represent in the applet?

    Select all that apply
    • A
    • B
    • C
    • D
    Check my answer (3)

    Slope dx/dy

    If the slope dy/dx=3.5 , what should the slope dx/dy be?

    Select all that apply
    • A
    • B
    • C
    • D
    Check my answer (3)

    Related Rates

    What is the relationship of t with the ratio of x and y changes in the applet?

    Select all that apply
    • A
    • B
    • C
    • D
    Check my answer (3)

    References/Acknowledgment:

    Special thanks to ChatGPT for contributing to and enhancing all of the sections of this GeoGebra applet based on my inputs.

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