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In-Depth Information
Figure 11.4
What is the hare's velocity at
t = 2.5 min
?
It's not immediately apparent how we might measure or calculate the
velocity at the exact moment t = 2.5, but observe that we can get a good
approximation by computing the average velocity of a very small interval
near t = 2.5. For a small enough interval, the graph is nearly the same
as a straight line segment, and the velocity is nearly constant, and so the
instantaneous velocity at any given instant within the interval will not be
too far off from the average velocity over the whole interval.
In
Figure 11.5,
we fix the left endpoint of a line segment at t = 2.5
and move the right endpoint closer and closer. As you can see, the shorter
the interval, the more the graph looks like a straight line, and the better
our approximation becomes. Thinking graphically, as the second endpoint
moves closer and closer to t = 2.5, the slope of the line between the end-
points will converge to the slope of the line that is tangent to the curve
at this point. A tangent line is the graphical equivalent of instantaneous
velocity, since it measures the slope of the curve just at that one point.
Let's carry out this experiment with some real numbers and see if we
cannot approximate the instantaneous velocity of the hare. In order to do
this, we'll need to be able to know the position of the hare at any given
time, so now would be a good time to tell you that the position of the hare
is given by the function
11
x(t) = −t
2
+ 6t − 1.
11
While it may seem nicely contrived that the hare's motion is described by a quadratic
equation with whole number coe
cients, we'll see later that it isn't as contrived as you
might think. Nature apparently likes quadratic equations. But you do have us on the
whole number coe
cients, which were cherry-picked.
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