I want to show pupils at an age of 14 to 17 and who do not like math what they can use analysis/analytic geometry in a job.
I could not remember an impressive example for most topics. Let's assume that pupils who are not convinced by maths yet will accept not to study physics or maths too. Hence I am looking for something like "A pilot calculates xx with the derivation of yy".
In which jobs do people use one of the following topics?
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One attention-getting application of exp() is compound interest. If you have time to develop the idea of compounding interest on money, then you might consider this route. The formula for compounding $n$ times a year with an APR of $r$ between $0$ and $1$ on an investment of $P(0)$ dollars is $P(t)=P(0)\cdot (1+\frac{r}{n})^{nt}$ where $t$ is in years. I always thought it was interesting that as $n$ increases you get more money, but it was a surprise to find out that there is an upper limit to what you will earn no matter how high $n$ goes!
For derivation, there is always something like the instantaneous speed of a falling object. A slightly different application that I like is actually more about differentials. I like the question about estimating the volume of paint needed to coat a big sphere using differentials. You can pick a more interesting shape or structure, of course.
For integration, you could outline the idea of finding volumes other than the ordinary objects they know the volume formulas for. If that seems too hard, maybe curve lengths.
Curves are so visual that you can probably sell curve sketching without an application. Find a lot of really cool graphs (especially polar graphs) and then apply a little calculus (without doing gritty details, though. Just say "if you were to go and find the critical points using the derivative, you would find these points. Let's find where these points are on the graph. Wow! This is exactly where this bend happens...") Here's a nice graph which I had the pleasure to talk about earlier.
I know you didn't mention probability, but integration and measure are applied in probablity theory. You can use integration to compute the probability a certain battery (or whatever other product) will last given the function of a probability distribution.
In general, if you're talking to a group about this, don't go down the road of going into too much detail. If the purpose is just to increase interest and intrigue, it's better not to do lengthy computations. If you manage to astound them by impressing upon them that math is interesting and not just a pile of numbers, you will have been a big success!
Actuaries use exp(x), derivation, and integration to find probabilities of risk in insurance policies. Engineers use math software to make sketches of surfaces and shapes in 3D. Physicists use integration and derivation to find work, mass using density functions, etc..