Too many exercises! (Calculus text with problems after each section *and* at the end of each chapter)

  • #1
Hello PF,
After thinking, googling, and reading many threads here, I decided that I'm going to start learning calculus using Stewart. The problem is that the book puts too many exercises after each section, in addition to those at the end of each chapter. What's the difference between these sets? I mean the exercise on each section and those at the end of each chapter. I don't have the time to solve all of them since I'm working, not just studying. Which of them should I solve?
Thanks in advance.
 

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  • #2
TeethWhitener
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Probably most folks here would tell you to solve as many exercises as possible. But I understand how trying to learn something while also having a whole host of other responsibilities can stretch someone thin.

I think if I had to choose which exercises to work on to maximize learning, I’d probably pick the ones I don’t immediately know how to do. It can be very satisfying to solve problems you already know how to solve, but ultimately, you don’t learn too much from it. But by going after problems that force you to think about how to solve them, you’ll probably learn more in the end, and you’ll work those problem-solving muscles in the process.
 
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  • #3
Probably most folks here would tell you to solve as many exercises as possible. But I understand how trying to learn something while also having a whole host of other responsibilities can stretch someone thin.

I think if I had to choose which exercises to work on to maximize learning, I’d probably pick the ones I don’t immediately know how to do. It can be very satisfying to solve problems you already know how to solve, but ultimately, you don’t learn too much from it. But by going after problems that force you to think about how to solve them, you’ll probably learn more in the end, and you’ll work those problem-solving muscles in the process.
Thanks a lot, that was very beneficial. I had an idea, which was: to go through the problem set made by MIT OCW for the corresponding course, which is way less in number and also probably chosen carefully by their professors. Like after finishing a chapter I solve its problem set given at OCW on that chapter. What do you think?
 
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  • #4
StoneTemplePython
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yes. This is the right approach -- mimic an existing 'good' syllabus.
 
  • #5
symbolipoint
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Hello PF,
After thinking, googling, and reading many threads here, I decided that I'm going to start learning calculus using Stewart. The problem is that the book puts too many exercises after each section, in addition to those at the end of each chapter. What's the difference between these sets? I mean the exercise on each section and those at the end of each chapter. I don't have the time to solve all of them since I'm working, not just studying. Which of them should I solve?
Thanks in advance.
You can choose which exercises to work through and the book supplies you with plenty. Think of this as an advantage. You studying on your own, or even if you were attending a class as enrolled student, will not need to solve EVERY EXERCSISE PROBLEM in every section , in every chapter. What you want is to choose a variety of exercises to help gain the best learning. The items at the end of the chapter are there to help you review ALL of that chapter.
 
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  • #6
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Pick one or two from each subtopic within the exercises. Then proceed. Usually, the more interesting problems are found near the end..
 
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  • #7
berkeman
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Usually, the more interesting problems are found near the end..
Way back when I was on my game in uni, I would work the textbook problems backwards in each chapter/section. Since the hardest problems would usually be at the end of the list, if I could solve the last few and they got less interesting/hard as I worked backwards, that meant that I had a good handle on that chapter/section. When the problems got less interesting and simpler, that was the cutoff point and time to move on to the next chapter/section.

If I had no clue on the last question or two in a chapter/section, that meant that I hadn't done a good job of learning the material...
 
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  • #8
mathwonk
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when i started teaching, i wanted to be prepared for any question, so thought i should solve every exercise. but when i tried that, i realized after a while that i had got the idea after doing the first half or so, and no longer needed to do the same type of thing over and over. so i suggest trying to develop a sense of when you have learned what they are trying to convey, and when you have done enough that you would be able to do the rest if needed. ie. do them until you feel confidence. but if you are still struggling with them, maybe you have not got it yet.
 
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  • #9
Vanadium 50
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If you solve more problems, you will learn more. So your question boils down to "I don't have time to learn everything. How much do I want to learn?"

I don't think we can answer that for you.

FWIW, I answered every single problem in Halliday & Resnick. Took a summer.
 
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  • #10
FWIW, I answered every single problem in Halliday & Resnick. Took a summer.
In your opinion, how long would it take to finish a book like Stewart's? like how many hours on average?
 
  • #11
Vanadium 50
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Never worked through Stewart,

H&R has maybe 1500 problems? A summer is 500 working hours? So that's 20 minutes per problem. That is very generous. Of course in real life what happens is that some problems take a few minutes and you spend the majority of your time on a minority of problems.

I would also point out that the solutions manual stinks. This is a good reason to say "no" to people asking for solutions manuals here. Sure, most of the time it's right, but there are plenty of errors. The solution to difficult problems is seldom enlightening, and in some cases they get the right answer by accident.
 
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  • #12
symbolipoint
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In your opinion, how long would it take to finish a book like Stewart's? like how many hours on average?
That is just the wrong question (maybe not more than just my opinion it is the wrong question) to ask. Would like want to be in danger of consuming all of the exercise questions in the book Would that be like, being in danger of consuming all of the exercises in the book?
 
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  • #13
symbolipoint
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Sure, most of the time it's right, but there are plenty of errors.
How do you mean, "errors"? Could those be not actually errors, but something else?
 
  • #14
symbolipoint
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In your opinion, how long would it take to finish a book like Stewart's? like how many hours on average?
Here's an idea. Let's say you exhausted ALL of the problems in your book so you have no more of them to do. (But unlikely as hell). Find an alternative, similar textbook and start working on the exercises in that one.
 
  • #15
berkeman
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That is just the wrong question (maybe not more than just my opinion it is the wrong question) to ask. Would like want to be in danger of consuming all of the exercise questions in the book?
Sorry, I'm not able to parse this. Even with single- and double-error correction turned on, my brain still isn't understanding it. Maybe try again? :smile:
 
  • #16
StoneTemplePython
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If you solve more problems, you will learn more. So your question boils down to "I don't have time to learn everything. How much do I want to learn?"
a better synopsis is OP has finite resources and tradeoffs are real. One option is to do everything in Stewart but there's e.g. a trade where OP does say an MIT syllabus of Stewart and then uses 'saved' time reading and solving problems in a more sophisticated text like say Spivak's Calculus or Abbott's analysis book. Many on this forum would say that's an obvious trade and that doing all the problems in Stewart is a dead-weight loss from this vantage (i.e. doing all of Stewart does not result in learning more in when opportunity costs are considered).

Based on prior posts (e.g. on investing), I was under the impression you do your thinking in terms of opportunity costs so I'm a little surprised here.

- - - - -
At a personal level two other things are:
1.) I always found it hard to learn from "calculus". Only after studying real (and complex) analysis did things make sense. So that's one bias.
2.) Philosophically I subscribe to something close to a Polya-ism which is a belief basically that it's better to solve an interesting (and challenging) problem several different ways than it is to solve several problems. Dumbing this down: emphasizing "quality" over "quantity".
 
  • #17
jtbell
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I think the large number of exercises in many textbooks is at least partly due to the fact that in the US university system, homework exercises often count for part of the final course grade.

Providing many exercises in textbooks allows instructors and professors to assign different sets of exercises in different years, and thereby reduce the effect of current students copying solutions from previous students.
 
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  • #18
symbolipoint
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Sorry, I'm not able to parse this. Even with single- and double-error correction turned on, my brain still isn't understanding it. Maybe try again? :smile:
I will make an adjustment to my post there; I see what the trouble is.
 
  • #19
PAllen
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I did very successful self study in high school (proven by tests and facility in later courses). My policy was simple. Most texts follow a few general rules:

- in line or section exercises most directly reflect the material just presented (occasionally, they provide an extension to the material, and thus are potentially challenging - wording and context indicates this).

- end of chapter exercises potentially use all prior material

- most of the time, exercises run from easier to harder, within any group of exercises

Thus, I simply did about half of all exercise of all the above types. I did not do back to front as suggested earlier. For me, it was better to solidify skills a bit with more basic exercises, then work to the harder ones. Given a more limited time budget, I would suggest starting with every third exercise, then doing more if time allows. But the fewer exercises provided, the more you should do. In books giving only a couple dozen per chapter, you better try them all. If you are trying to learn from a book without exercises, try really hard to formulate your own scenarios similar to, but not the same as anything covered in the text, and work them out (or, perhaps better, try to find online exercise collections for the topic; but that option didn't exist, back in the day).
 
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  • #20
I would suggest starting with every third exercise, then doing more if time allows.
Thanks for the very useful answer. But, what did you mean by starting with every third exercise? Just to make sure that I understood :)
 
  • #21
PAllen
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Thanks for the very useful answer. But, what did you mean by starting with every third exercise? Just to make sure that I understood :)
I mean doing exercise 1, 4, 7, 10, etc. in any group.
 
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  • #22
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