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2. Originally Posted by SameOldThing
For my calculus final project we are assigned a topic and are supposed to explain how it has helped us in calculus so far. My topic is differentiability which is very broad. Our textbook separates this sections into: how f'(a) might fail to exist, Differentiability implies local linearity, derivatives on a calculator, differentiability implies continuity, and the intermediate value theorem for derivatives.

I have already made a powerpoint explaining the topics themselves and I need help explaining how each section helps further.
Hmm, I don't quite understand the "explain how it has helped us in calculus so far" part. That's like saying, explain how learning calculus has helped you learn calculus...

But it seems you just need to give some examples that involve differentiability? So, example functions, or example word problems. One example that's probably not in your textbook is ray tracing (which has real world applications). So, to find how a ray of light bounces off an object, you can use derivatives to get the tangent line, hence the normal, hence you can get the reflected ray. And you can have your object be non-differentiable in some parts. (In practice for ray tracing, we could be using more advanced techniques from vector calculus/linear algebra, but it's the same idea).

Another example is using Newton's method to find square roots ( $f(x)=\sqrt{x}$ is differentiable, and thus locally linear).