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[music].
Welcome back to Calculus One, and welcome

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to week eight of our time together.
Last week, during week seven, we began

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studying applications of the derivative
and we focused in on related rates

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problems.
It's not so surprising, I think that

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calculus can solve these related rates
problems.

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After all calculus is all about how things
change, so it makes sense that you could

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use calculus to study how things are
changing together.

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This week, in week eight, we focus on
optimization problems.

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Optimization problems are all about the
best of something, right?

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You want to find the shape of a soup can
that maximizes the amount of soup for a

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given amount of metal.
Or you want to find the shape of

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rectangular fence that encloses the most
area for a given length of fence.

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Or you want to find the shape of
rectangular fence that encloses the most

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area for a given length of fence.
Or you want to understand why bubbles form

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the shapes that they do or why light
travels on the paths that it does.

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Right?
All of these are problems about finding

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the best of something and that's really
different in related rates problems.

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Right?
We're not just asking about how something

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changes, I'm actually trying to find a
point where some function achieves a

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maximum or a minimum value.
And I think it is really surprising that

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calculus, which fundamentally, is all
about how things change can also be used

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to solve that problem.
It can also be used to find a specific

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value or something really exiting happens.
I also want to say thank you.

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It's is a long and difficult course, and
you've made it now, more than half way

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through.
Congratulations.