Consider an automatic door. At time t=0 seconds, someone steps on a pad. The door swings open, slows down, stops momentarily and then starts closing. At any point in time we can measure the angle the door makes with what would be the closed position. So at t=0, d=0.
We can model this by looking at the graph of d(t)=200t * 2^{-t}.
var('t')
d(t)=200*t*2^(-t)
A=plot(d,(t,0,10))
show(A)
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An amount function which is locally linear has an associated rate function. Using calculus, we refer to the rate function as the derivative.
In this case d(t) can be considered an amount function. The curve shown in red is the rate function for d(t). On paper, we refer to this as d'(t), which is pronounced "d prime of t."
R=plot(d.derivative(t),(t,0,10), color="red")
show(R)
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Take a look at both graphs on the same set of axes.
- When the amount graph is increasing, what is happening to the rate graph?
- When the amount graph is decreasing, what is happening to the rate graph?
- Why does this make sense?
show(A+R)
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