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2: Average rates of change, average velocity and the secant line

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    121085

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    2.1.png
    Figure 2.1: Cell biologists track the motion of proteins inside cells on images like this kymograph. Data courtesy of Anna Akhmanova’s lab shows the position of proteins that track growing ends of long biopolymers called microtubules as they get longer or shrink inside cells. Displacement is shown horizontally (scale bar \(5 \mu \mathrm{m}\) ) and time vertically (scale bar \(20 \mathrm{sec}\) ).

    A physicist might study the motion of a falling ball by taking strobe images at fixed time intervals, and gluing them side by side to get a record of position of the ball over time. In a similar manner, cell biologists study the motion of proteins inside living cells. First, the proteins are labeled by fluorescent "tags" (this makes them visible in microscopic images). Then images of some thin strip of the cell are made at fixed time intervals, in regions through which the "glowing" (fluorescent) proteins move. Finally, those thin strips are "glued" together to form a record of the protein position over time, as shown in each panel of Fig 2.1. Biologists refer to such images as kymographs.

    The "streaks of light" in such kymographs allow us to determine the locations of the labeled proteins over time. as well as their velocity in the cell. But how fast were these proteins moving? Why are there zigzags in the left panel? And what happened in the treated cells (right panel) that made the streaks look different from those in the "normal cell" (left panel) \({ }^{1}\) ?

    In this chapter we develop the tools to address some of these questions, and to characterize what we mean by velocity. As a first step, we introduce average rate of change. To motivate the idea, we examine data for common processes: changes in temperature, and motion of a falling object. Simple experiments are described in each case, and some features of the data are discussed. Based on each example, we calculate net change over some time interval and then define the average rate of change and average velocity. This concept generalizes to functions of any variable (not only time). We interpret this idea geometrically, in terms of the slope of a secant line.

    In both cases, we ask how to use average rate of change (over a given interval) to find better and better approximations of the rate of change at a single instant, (i.e. at a point). We find that one way to arrive at this abstract concept entails refining the dataset - collecting data at closer and closer time points. A second - more abstract - way is to use a limit. Eventually, this procedure allows us to arrive at the definition of the derivative, which is the instantaneous rate of change.

    \({ }^{1}\) Benjamin P Bouchet, Ivar Noordstra, Miranda van Amersfoort, Eugene A Katrukha, York-Christoph Ammon, Natalie D Ter Hoeve, Louis Hodgson, Marileen Dogterom, Patrick WB Derksen, and Anna Akhmanova. Mesenchymal cell invasion requires cooperative regulation of persistent microtubule growth by slain 2 and clasp1. Developmental cell, 39(6):708-723, 2016

    This page titled 2: Average rates of change, average velocity and the secant line 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.