What we can do is approximate the tangent line by picking two points
on our graph and computing the slope of the line between them.
Lets compute the slope of the line through the points (2, f(2)) and (3, f(3)).
f(2) = 16 · 22 = 64 and f(3) = 16 · 33 = 144
We can see that the line with slope 80 passing through (2, 64) is not the
tangent line, and does not have the correct slope, but we can find better
approximations by picking our second point closer to (2, 64). So lets pick our
second point to be (2.5, f(2.5).
f(2.5) = 16 · (2.5)2 = 100
We could continue to pick values closer to 2, and each would be a better
approximation. To get the exact answer seems to be an impossible task, but
in the 17th century the Mathematician and Physicist , Sir Isaac Newton had
an idea. Instead of picking the second point before you calculate the slope ,
pick the second point after you compute the slope. The way he did that
was to pick the second time to be an increment h bigger than the first thus
the second point would be (2+h, f(2+h) The value h would be de termined
after we do our computations. In our example we have
So the object strikes the ground with velocity 25 meters per second in
a downward direction.
4. To find the velocity of the object at its maximum height we compute
its velocity at the time it is at its maximum height which we know to
be seconds. We also know that height to be
meters, so we must
compute the height at seconds.
So we have
So we conclude the object has 0 velocity at its maximum height. But
this should be obvious to us, since it has stopped going up and is about
to start going down.
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