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3: Three Faces of the Derivative - Geometric, Analytic, and Computational

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    121093

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    In Chapter 2 we bridged two concepts: the average rate of change (slope of secant line) and the instantaneous rate of change (the derivative). We arrived at a technique for calculating a derivative algebraically. As a result, we introduced limits - a concept that merits further discussion. One goal of this chapter is to consider the technical aspects of limits - a requirement if we are to use the definition of the derivative to determine derivatives of common functions.

    But first, we consider a distinct approach which is geometric in flavour. Namely, we show that the local behavior of a continuous function is described by a tangent line at a point on its graph: we can visualize the tangent line by zooming into the graph of the function. This duality - the geometric (graphical) and analytic (algebraic calculation) views - form themes throughout the discussions to follow. They are two complementary, but closely related approaches to calculus.

    • 3.1: The Geometric View - Zooming into the Graph of a Function
      In this section we consider well-behaved functions whose graphs are "smooth", as opposed to the discrete data points of Chapter 2. We link the derivative to the local shape of the graph of the function. By local behavior we mean the shape we see when we zoom into a point on the graph. Imagine using a microscope where the center of the field of vision is some point of interest. As we zoom in, the graph looks flatter, until we observe a straight line.
    • 3.2: The Analytic View - Calculating the Derivative
      We now make the intuitive discussion more precise with a formal definition, based on the concept of a limit. We first define what it means for a function to be continuous, and then show how limits are computed to test that definition.
    • 3.3: The Computational View - Software to the Rescue!
      We have explored geometric and analytic aspects of the derivative. Here we show a third aspect of the derivative: its numerical implementation using a simple spreadsheet. The ideas introduced here reappear in a variety of problems where repetitive calculations are needed to arrive at a solution.
    • 3.4: Summary
    • 3.5: Exercises

    This page titled 3: Three Faces of the Derivative - Geometric, Analytic, and Computational is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Leah Edelstein-Keshet via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.