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Fundamental theorem of Calculus failure cause [closed]
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I know the examples don’t of trying it with improper integrals don’t give sensible answers, and why they don’t make sense, but what makes them not work? What is it in the geometric, word/thought experiment, graphical, or anything else intuition that comes from our definition of integrals that makes us have to, say, split it at a singularity? Surely numerical integration only gives you one result, but splitting it like this gives you two. Is there something that’s leads us to knowing that it can fail without having to try examples? Just some context. Any insight is appreciated!
calculus
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
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closed as unclear what you're asking by user7530, RRL, John Douma, Kavi Rama Murthy, Kemono Chen Jan 24 at 2:48
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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show 2 more comments
$begingroup$
I know the examples don’t of trying it with improper integrals don’t give sensible answers, and why they don’t make sense, but what makes them not work? What is it in the geometric, word/thought experiment, graphical, or anything else intuition that comes from our definition of integrals that makes us have to, say, split it at a singularity? Surely numerical integration only gives you one result, but splitting it like this gives you two. Is there something that’s leads us to knowing that it can fail without having to try examples? Just some context. Any insight is appreciated!
calculus
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
$endgroup$
closed as unclear what you're asking by user7530, RRL, John Douma, Kavi Rama Murthy, Kemono Chen Jan 24 at 2:48
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
Can you give a more specific problem?
$endgroup$
– Math Lover
Jan 23 at 22:29
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@MathLover I had -1 to 1 of $frac{1}{x^2}$ on my mind but I don’t really care about it, although I guess that’s an example if you want one. Computation thus far hasn’t helped me understand in depth so I doubt some specific evaluation will help.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:40
$begingroup$
This question is looking for something specific, but it’s not a particular problem. I can’t connect having to solve improper integrals differently to the foundational concepts of derivatives and integrals. That is, when I do calc it usually feels like what I’m writing is a just a representation of something grounded and intuitive happening in my head but this is not the case here.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:48
$begingroup$
I think integrals are just the way they are, so if there is some things that doesn’t work, you can just check if the steps proving this doesn’t work for certain numbers-I think, but I’m not that sure.
$endgroup$
– Math Lover
Jan 23 at 22:50
1
$begingroup$
If the hypotheses of the theorem aren't satisfied, you can hardly expect the theorem to hold.
$endgroup$
– saulspatz
Jan 23 at 23:26
|
show 2 more comments
$begingroup$
I know the examples don’t of trying it with improper integrals don’t give sensible answers, and why they don’t make sense, but what makes them not work? What is it in the geometric, word/thought experiment, graphical, or anything else intuition that comes from our definition of integrals that makes us have to, say, split it at a singularity? Surely numerical integration only gives you one result, but splitting it like this gives you two. Is there something that’s leads us to knowing that it can fail without having to try examples? Just some context. Any insight is appreciated!
calculus
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
$endgroup$
I know the examples don’t of trying it with improper integrals don’t give sensible answers, and why they don’t make sense, but what makes them not work? What is it in the geometric, word/thought experiment, graphical, or anything else intuition that comes from our definition of integrals that makes us have to, say, split it at a singularity? Surely numerical integration only gives you one result, but splitting it like this gives you two. Is there something that’s leads us to knowing that it can fail without having to try examples? Just some context. Any insight is appreciated!
calculus
calculus
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
asked Jan 23 at 22:22
Benjamin ThoburnBenjamin Thoburn
354313
354313
closed as unclear what you're asking by user7530, RRL, John Douma, Kavi Rama Murthy, Kemono Chen Jan 24 at 2:48
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by user7530, RRL, John Douma, Kavi Rama Murthy, Kemono Chen Jan 24 at 2:48
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
$begingroup$
Can you give a more specific problem?
$endgroup$
– Math Lover
Jan 23 at 22:29
$begingroup$
@MathLover I had -1 to 1 of $frac{1}{x^2}$ on my mind but I don’t really care about it, although I guess that’s an example if you want one. Computation thus far hasn’t helped me understand in depth so I doubt some specific evaluation will help.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:40
$begingroup$
This question is looking for something specific, but it’s not a particular problem. I can’t connect having to solve improper integrals differently to the foundational concepts of derivatives and integrals. That is, when I do calc it usually feels like what I’m writing is a just a representation of something grounded and intuitive happening in my head but this is not the case here.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:48
$begingroup$
I think integrals are just the way they are, so if there is some things that doesn’t work, you can just check if the steps proving this doesn’t work for certain numbers-I think, but I’m not that sure.
$endgroup$
– Math Lover
Jan 23 at 22:50
1
$begingroup$
If the hypotheses of the theorem aren't satisfied, you can hardly expect the theorem to hold.
$endgroup$
– saulspatz
Jan 23 at 23:26
|
show 2 more comments
1
$begingroup$
Can you give a more specific problem?
$endgroup$
– Math Lover
Jan 23 at 22:29
$begingroup$
@MathLover I had -1 to 1 of $frac{1}{x^2}$ on my mind but I don’t really care about it, although I guess that’s an example if you want one. Computation thus far hasn’t helped me understand in depth so I doubt some specific evaluation will help.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:40
$begingroup$
This question is looking for something specific, but it’s not a particular problem. I can’t connect having to solve improper integrals differently to the foundational concepts of derivatives and integrals. That is, when I do calc it usually feels like what I’m writing is a just a representation of something grounded and intuitive happening in my head but this is not the case here.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:48
$begingroup$
I think integrals are just the way they are, so if there is some things that doesn’t work, you can just check if the steps proving this doesn’t work for certain numbers-I think, but I’m not that sure.
$endgroup$
– Math Lover
Jan 23 at 22:50
1
$begingroup$
If the hypotheses of the theorem aren't satisfied, you can hardly expect the theorem to hold.
$endgroup$
– saulspatz
Jan 23 at 23:26
1
1
$begingroup$
Can you give a more specific problem?
$endgroup$
– Math Lover
Jan 23 at 22:29
$begingroup$
Can you give a more specific problem?
$endgroup$
– Math Lover
Jan 23 at 22:29
$begingroup$
@MathLover I had -1 to 1 of $frac{1}{x^2}$ on my mind but I don’t really care about it, although I guess that’s an example if you want one. Computation thus far hasn’t helped me understand in depth so I doubt some specific evaluation will help.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:40
$begingroup$
@MathLover I had -1 to 1 of $frac{1}{x^2}$ on my mind but I don’t really care about it, although I guess that’s an example if you want one. Computation thus far hasn’t helped me understand in depth so I doubt some specific evaluation will help.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:40
$begingroup$
This question is looking for something specific, but it’s not a particular problem. I can’t connect having to solve improper integrals differently to the foundational concepts of derivatives and integrals. That is, when I do calc it usually feels like what I’m writing is a just a representation of something grounded and intuitive happening in my head but this is not the case here.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:48
$begingroup$
This question is looking for something specific, but it’s not a particular problem. I can’t connect having to solve improper integrals differently to the foundational concepts of derivatives and integrals. That is, when I do calc it usually feels like what I’m writing is a just a representation of something grounded and intuitive happening in my head but this is not the case here.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:48
$begingroup$
I think integrals are just the way they are, so if there is some things that doesn’t work, you can just check if the steps proving this doesn’t work for certain numbers-I think, but I’m not that sure.
$endgroup$
– Math Lover
Jan 23 at 22:50
$begingroup$
I think integrals are just the way they are, so if there is some things that doesn’t work, you can just check if the steps proving this doesn’t work for certain numbers-I think, but I’m not that sure.
$endgroup$
– Math Lover
Jan 23 at 22:50
1
1
$begingroup$
If the hypotheses of the theorem aren't satisfied, you can hardly expect the theorem to hold.
$endgroup$
– saulspatz
Jan 23 at 23:26
$begingroup$
If the hypotheses of the theorem aren't satisfied, you can hardly expect the theorem to hold.
$endgroup$
– saulspatz
Jan 23 at 23:26
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Think of the fundamental theorem this way. If you periodically measure the speed at which the distance of an object from given point is changing, you can estimate the distance of the object from that point. That is, $f(t)-f(0)approx sumtriangle x_n f'(x_n)$ The (second) fundamental theorem of calculus says that this approximation becomes an equality in the limit.
Think of a ship in the days of sail, before there were good star charts. They figured out approximately where they were, in part, by measuring their speed though the water every so often ("casting the log.") But now suppose the Sea God picks the ship up and puts it down somewhere else instantaneously, so that the ship's velocity is undefined. Measuring is pointless now.
I hope this is intuitive enough.
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Think of the fundamental theorem this way. If you periodically measure the speed at which the distance of an object from given point is changing, you can estimate the distance of the object from that point. That is, $f(t)-f(0)approx sumtriangle x_n f'(x_n)$ The (second) fundamental theorem of calculus says that this approximation becomes an equality in the limit.
Think of a ship in the days of sail, before there were good star charts. They figured out approximately where they were, in part, by measuring their speed though the water every so often ("casting the log.") But now suppose the Sea God picks the ship up and puts it down somewhere else instantaneously, so that the ship's velocity is undefined. Measuring is pointless now.
I hope this is intuitive enough.
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
$endgroup$
add a comment |
$begingroup$
Think of the fundamental theorem this way. If you periodically measure the speed at which the distance of an object from given point is changing, you can estimate the distance of the object from that point. That is, $f(t)-f(0)approx sumtriangle x_n f'(x_n)$ The (second) fundamental theorem of calculus says that this approximation becomes an equality in the limit.
Think of a ship in the days of sail, before there were good star charts. They figured out approximately where they were, in part, by measuring their speed though the water every so often ("casting the log.") But now suppose the Sea God picks the ship up and puts it down somewhere else instantaneously, so that the ship's velocity is undefined. Measuring is pointless now.
I hope this is intuitive enough.
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
$endgroup$
add a comment |
$begingroup$
Think of the fundamental theorem this way. If you periodically measure the speed at which the distance of an object from given point is changing, you can estimate the distance of the object from that point. That is, $f(t)-f(0)approx sumtriangle x_n f'(x_n)$ The (second) fundamental theorem of calculus says that this approximation becomes an equality in the limit.
Think of a ship in the days of sail, before there were good star charts. They figured out approximately where they were, in part, by measuring their speed though the water every so often ("casting the log.") But now suppose the Sea God picks the ship up and puts it down somewhere else instantaneously, so that the ship's velocity is undefined. Measuring is pointless now.
I hope this is intuitive enough.
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
$endgroup$
Think of the fundamental theorem this way. If you periodically measure the speed at which the distance of an object from given point is changing, you can estimate the distance of the object from that point. That is, $f(t)-f(0)approx sumtriangle x_n f'(x_n)$ The (second) fundamental theorem of calculus says that this approximation becomes an equality in the limit.
Think of a ship in the days of sail, before there were good star charts. They figured out approximately where they were, in part, by measuring their speed though the water every so often ("casting the log.") But now suppose the Sea God picks the ship up and puts it down somewhere else instantaneously, so that the ship's velocity is undefined. Measuring is pointless now.
I hope this is intuitive enough.
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
edited Jan 24 at 12:44
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
answered Jan 23 at 23:46
saulspatzsaulspatz
17.2k31435
17.2k31435
add a comment |
add a comment |
1
$begingroup$
Can you give a more specific problem?
$endgroup$
– Math Lover
Jan 23 at 22:29
$begingroup$
@MathLover I had -1 to 1 of $frac{1}{x^2}$ on my mind but I don’t really care about it, although I guess that’s an example if you want one. Computation thus far hasn’t helped me understand in depth so I doubt some specific evaluation will help.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:40
$begingroup$
This question is looking for something specific, but it’s not a particular problem. I can’t connect having to solve improper integrals differently to the foundational concepts of derivatives and integrals. That is, when I do calc it usually feels like what I’m writing is a just a representation of something grounded and intuitive happening in my head but this is not the case here.
$endgroup$
– Benjamin Thoburn
Jan 23 at 22:48
$begingroup$
I think integrals are just the way they are, so if there is some things that doesn’t work, you can just check if the steps proving this doesn’t work for certain numbers-I think, but I’m not that sure.
$endgroup$
– Math Lover
Jan 23 at 22:50
1
$begingroup$
If the hypotheses of the theorem aren't satisfied, you can hardly expect the theorem to hold.
$endgroup$
– saulspatz
Jan 23 at 23:26