I have a programming background and have recently got quite interested in electronics and signal processing. After picking up just about every book/resource on those two subjects, it's pretty clear how important maths is. Learning maths is quite an investment though, especially to the level my electronic engineering buddies have learned through university.
I was wondering if there's any point in picking up a calculus book and instead of working problems by hand I work them using Matlab/Maple/ (probably a free/open-source one)? How bad/good of an idea is this? Can it even work? Will I know what to do when it comes to solving those troublesome "word" problems?
I've found myself in a similar position before, coming from a comp sci + applied math background, but suddenly needing some signal processing knowledge.
Personally, I think it is more important to understand the things conceptually. Even if you can (should, really) take some Fourier transforms by hand, if you do not understand why your physicist colleague is interested in peaks in the power spectrum or what correcting image blur has to do with Fourier analysis, then you will have more important problems.
I'd suggest just reading a book and doing the non-rote, conceptual questions (i.e. the thinking ones). I am quite sure that most problems you will face in signal processing will not be analytically tractable anyway, so numerics and concepts are more useful.
I think Mathematica is great for checking results in rote work, and matlab is ok for visualizing numerical things easily, but neither is helpful in learning concepts.