Good Book Of Math Problems / On Problem Solving

[moe darklight]

I'm looking for a good book with problems that require a creative approach, or on problem solving in general. Questions in my textbooks usually cover exactly what the chapter covered, and require little more than plugging numbers. There are maybe 2 or three problems in each chapter that require a little more thought, and sometimes I miss obvious answers; I feel this is because I don't get enough practice at approaching tougher problems. I figure the best way to hone this skill is practice.

I'm not looking for anything insanely tough here just a way to apply and connect what I'm learning in a different way. ... I'm first year first semester and have taken Calculus and Linear Algebra I, so it should be at that level.

Thanks

 

[mathwonk]

as usual, the book by spivak, calculus, has many challenging problems. on general problem solving, the classic is "how to solve it" by the late great george polya of stanford. working through my online free linear algebra notes, the 15 page version, should serve as a good exercise for you, just filling in the gaps and doing the exercises, since you have already had the course of linear algebra.

here's a problem for you to start on: prove that if f is a differentiable function on an interval containing a<b, and if f'(a) < 0, and f'(b) > 0, then there is a point c between a and b where f'(c) =0. do NOT assume f' is continuous.

there are also contest books, e.g. from russia, with good problems. also in any book, trying to prove the theorems in that book without looking at the proofs given there is a superb exercise. I do it all the time in teaching nowdays, since otherwise if i read the stupid explanation in the lousy book we use i do not understand anything. but if i find a proof myself i understand it well and also can explain how i thought of it.

 

[moe darklight]

also in any book, trying to prove the theorems in that book without looking at the proofs given there is a superb exercise.

Yes, sometimes I try to do that. And while I'm reading the chapter I work out the examples and then continue reading to the explanation.
This semester has been a bit weird because the faculty went on a 3-week strike at my school, so we've been on sort of overdrive to catch up, so I haven't had as much time as I'd like to spend on any of the subjects.