Rates of Change (The 2nd Pillar of Calculus)

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Rates of Change

Tracking change
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Definition of derivative

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Related Rates

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The Fundamental Theorem of Calculus

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Exploring average velocity

Before we start talking about instantaneous rate of change, let's talk about average rate of change. A simple example is average velocity. If you drive 180 miles in 3 hours, then your average speed is 60 mph. We get this by dividing the distance traveled by the time:

$v_{avg} = \frac{\Delta s}{\Delta t},$ where

$\Delta s$ is the distance traveled and

$\Delta t$ is the time elapsed. We use the Greek letter

$\Delta$ to mean "change in". If we start at position

$s(t_0)$ at time

$t_0$ and end up at position

$s(t_1)$ at time

$t_1$ , then

$\begin{eqnarray*} \Delta s & = & s(t_1) - s(t_0), \cr \Delta t & = & t_1 - t_0. \end{eqnarray*}$ The following video goes over these ideas, and compares them to the slope of a secant line.