I (a student) am doing a presentation on Taylor series in my class (12th grade, in Germany if this is relevant). I am looking for a good example where you can see when Taylor series might be useful. Something like
Consider this problem with this function. [Some information on how the problem is hard to solve with its current function]. Wouldn't it be practical to have a "polynomial version" of this function to solve the problem? [Some information on why this would make the problem easier]
Of course, I would be open to other examples/applications that an average 12th-grade student would understand.
I looked at What are the practical applications of the Taylor Series? on Math SE, but none of the applications/examples there seem to be showing the practicability of Taylor series. For example, we (unfortunately) haven't done differential equations.
In his video Taylor series | Chapter 11, Essence of calculus (which basically is how I learnt Taylor series), 3blue1brown (Grant Sanderson) tells how he was first confronted with Taylor series:
[...] this cosine function made the problem awkward and unwieldly. But by approximating [using Taylor series], everything fell into place much more easily.
It isn't explained how exactly the approximation made things easier, but I think this is the sort of example/application I'm looking for for my introduction – some problem, for example, where you would easily see that a polynomial approximation is a lot better to work with.