livingcalculus.com

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  • Introducing the Chain Rule, Part 2: Connecting Composition and Related Rates

    One reason that so many students struggle with the chain rule is that they never really got the point of composition in the first place. Sure they can “put one function inside another,” but they are missing this essential point:

    Composite functions appear in situations where the value of one quantity is determined by another, which in turn is determined by another.

    Functions appear in situations where the value of one quantity is determined by the value of another.

    Understanding composition this way builds a natural bridge between related rates and the idea of composition. For example, consider the following scenarios:

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  • About the Cover of Living Calculus

    I painted the opening flower on the cover to be a visual representation of growth and change. It is a symbol of calculus as a vibrant and living field, and also a symbol of the dynamic world that calculus helps us understand.

    The image in the lower-right shows the Indian astronomer Aryabhata, the first person known to have taken a derivative, looking up at a modern representation of his work with the sine function. The image was produced in Adobe Illustrator, based on a photograph of a sculpture of Aryabhata.

    The graphs in the lower left illustrate the connection between the volume and flow rate through Lake Granby, a reservoir in Colorado. They represent the important role calculus can play in understanding modern issues such as our changing water supply.

    If you like this and want a copy of my whole book, please click here