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I'm currently in a class that uses Spivak's Calculus. Because of (fairly) good explanations and proofs, fun and challenging exercises, and some witty comments he gives occasionally, I really enjoy using this book. The only problem I find, however, is that sometimes I spend far too long on some problems - which means that I'm not able to cover all problems before a test/exam. Strictly in interest of time, is it justified to use the solution manual to go over the solutions to SOME problems (and then reproduce the proofs myself without looking again), or should I work hard on each and every problem that I have the time for (knowing that doing this probably won't allow me to finish all)? Which one is better for gaining mastery of the content?

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    $\begingroup$ I think if you have spent, say, an hour's effort on a problem, it is justified to check the answer. Perhaps just enough lines of it to get you back on the right track, then complete the proof yourself for the rest? It's impossible to do every problem - don't beat yourself up for not doing so. Ask your professors - I doubt they ever did every problem in the textbooks themselves. You have to be selective. $\endgroup$ – bounceback Jan 25 at 22:16
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    $\begingroup$ I’d say that it depends on what kind of mastery you aspire to achieve. Being stubborn certainly helps with developing an own style and engaging one’s creativity, but if you want to merely study the contents of mathematics, you will be more efficient if you look up the solutions to the questions you feel you are taking too long to solve. You nevertheless shouldn’t make looing up solutions a habit, that would just lower your threshold of frustration in the long run. If you do both, you will develop a sense for when you’re really stuck and when you’re merely too tired or lazy. I say, experiment … $\endgroup$ – k.stm Jan 25 at 22:18
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This is definitely a question many students face, and has been discussed a bit on this site as well (see this excellent answer). In general, I think it's unwise to try and study and completely understand/come up with a solution to each and every exercise in a book. You should definitely read the solution after struggling for a while (~$1$h) and (re)reading the text, thinking about the exercises. But there might be literally thousands of exercises, and it would take significant effort to come up with all their solutions yourself, hence I think you should not attempt every question, and even in those you do, give up after a while if you cannot solve them. Instead, in textbooks which explicitly mark difficulty level, I would recommend trying $5$ or so from the "basic" questions per section/chapter to make sure you get the basics, $2$ to $3$ from the middle-level question, and $1$ question from the most difficult level. You should see the linked answer for more in depth discussion.

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    $\begingroup$ Also if you do them all, you won't have any left over for when you want to practise some in a few months' time! I think there possibly need to be "too many" questions in the textbook, for that reason. $\endgroup$ – timtfj Jan 26 at 0:04

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