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In this unit, we have seen the derivative appear in several guises.
- As the slope of a graph when viewed under a very powerful microscope.
- As the instantaneous rate of change of some
physical process.
- If a function repents the position of an object as a function
of time, then the function's derivative represents the instantaneous
velocity of the object.
It can be confusing to see the same idea from so many different
angles (remember the blind men touching the elephant?). If so, don't
worry too much because the rest of the course is intended to explore
these ideas more deeply. It will help you to see, as we will shortly,
what the derivative is used for. Gradually, you will be able to move
back and forth between these seemingly different interpretations with
ease. What you should recognize in all three scenarios is that the
derivative is measuring how a function changes over a very small scale
(infinitesimally small, in fact).
You might want to explore a bit further and see what other folks
in physics and mathematics courses have to say about this and related
examples. The links below will help you get started. You might also
want to search the web for the topics "projectile motion", "falling
ball" and related concepts that we have discussed here.
Other sites to visit
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Note: The falling Ball is an example of what is called 1 dimensional motion
in mechanics. It is related to more complicated motions (such as the
motion of a thrown ball in 2 and 3 dimensions). We will not discuss these more
complicated motions now, but you may want to peak at what other people
have to say about these by visiting:
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So what is the derivative, after all?
