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Arc Length: $L = \theta r$
Again, this is an example of a linear relationship, but it's called "bi-linear", because it's linear in two different ways: $L$ is proportional to each of $\theta$ and $r$. So
Now you might wonder if there should be, or could be, an extra constant in there, but you can see that the formula works out for the circumference formula for a circle: \[ \theta=2\pi, \textrm{ so } L=2 \pi r \]
Yep! That's the formula for the circumference of a circle.
We can use this diagram (and the arc length formula) to prove a really important limit: \[ \lim\limits_{\theta \to 0}{\frac{\sin(\theta)}{\theta}}=1 \]
(See page 142, which I'll walk you through now. The proof uses some elementary trigonometry, plus the squeeze theorem, which is one of the really cool things about this proof!)
We will derive the derivative of the sine function from the limit definition of the derivative (although we can see the derivative graphically above).
There are exactly three important trig identities one needs to know (all the others can be derived from these three):
So let's see how to derive the derivative of the sine from the MIDIC -- the Most Important Definition In Calculus -- the limit definition of the derivative -- using the second of these identities, plus one more important limit (which can be proven with conjugates, and the first limit we derived): \[ \lim\limits_{\theta \to 0}{\frac{\cos(\theta)-1}{\theta}}=0 \]
Then the derivative of the cosine can be derived by simply shifting the sine function, and using the second trig identity above.
I personally think this to myself: "f prime of stuff times the derivative of the stuff"; or "f prime of stuff times stuff prime.",
where $f$ is the "outside" function:
You can see that the rule is fairly simple, once you've identified the composition -- that is, once you've torn apart $F$ to find $f$ and $g$.
You might take a look at this summary from my pre-calc class to review compositions.
Before we do that, however, I'd like to show how to derive the chain rule, using the limit definition of the derivative. Everything comes from that! I'll need to use a result from p. 153, where the chain rule is proved in our text.
Basically, however, it relies on the tangent line: we want to use the fact that
Why does that make sense? Because it comes straight out of the limit definition, where we throw away the limit. That's why we have to write "$\approx"$:
If time were measured in years from January, would
be a good model? What would be a good choice for the parameter $A$?