Might be we have different chunking. In my preconceptions calculus is the first necessary stepping stone to the other stuff you mentioned. I have no idea how to approach Fourier transform conceptually for example than by the calculus route since the integral form is always introduced first. It's true linear algebra and calculus don't often meet at first - until one needs to do coordinate tranforms from e.g. spherical coordinates to cartesian.
It's true I don't need that suff in my daily work that much. But I recognise a lot of problems I might meet are trivial with some applied calculus. Like the newton iteration, which you mentioned.
http://www.dspguide.com/ch8/1.htm talks about the discrete Fourier transform, which decomposes a discrete (periodic) signal into a sum of a discrete set of sinusoids. The Fourier transform is actually a case where the continuous case is misleading — in the continuous case, you unavoidably have the Gibbs phenomenon, a complication which disappears completely in the discrete case, and the argument for this is a great deal simpler than the analogous reasoning for analytic signals. And even if you show that, for example, sinusoids of different frequencies are orthogonal in the continuous case, it doesn't immediately follow that this is true of the sampled versions of those same signals — and in fact it isn't true in general, only in some special cases. You can show by a simple counting argument that no other sampled sinusoids are orthogonal to the basis functions of a DFT, for example. Showing that the DFT basis is orthogonal is more difficult!
You definitely don't need calculus to transform between spherical and Cartesian coordinates. I mean I'm pretty sure Descartes did that half a century before calculus was invented. You do need trigonometry, which is about a thousand years older.
Newton iteration is a bit dangerous; it can give you an arbitrary answer, and it may not converge. In cases where you think you might need Newton iteration, I'd like to suggest that you try interval arithmetic (see http://canonical.org/~kragen/sw/aspmisc/intervalgraph), which is guaranteed to converge and will give you all the answers but is too slow in high dimensionality, or gradient descent, which does kind of require that you know calculus to understand and works in more cases than Newton iteration, although more slowly.
It's true I don't need that suff in my daily work that much. But I recognise a lot of problems I might meet are trivial with some applied calculus. Like the newton iteration, which you mentioned.