https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FFundamental_theorem_of_calculusUser:IssaRice/Fundamental theorem of calculus - Revision history2026-05-26T19:02:26ZRevision history for this page on the wikiMediaWiki 1.41.2https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3404&oldid=prevIssaRice at 18:26, 8 September 20212021-09-08T18:26:10Z<p></p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area. [PROOF? -- <del style="font-weight: bold; text-decoration: none;">hmm, a </del>rigorous proof of this <del style="font-weight: bold; text-decoration: none;">''might'' just require </del>FTC2, which makes this visualization circular... but you can think of it intuitively too, like<del style="font-weight: bold; text-decoration: none;">... </del>as long as F has no jumps, you can get nearby values of F by adding lots of or little bits of area.]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area. [PROOF? -- <ins style="font-weight: bold; text-decoration: none;">an easy </ins>rigorous proof of this <ins style="font-weight: bold; text-decoration: none;">uses </ins>FTC2, which makes this visualization circular... but you can <ins style="font-weight: bold; text-decoration: none;">also prove this using the fact that every antiderivative is a constant away, which doesn't require FTC. </ins>think of it intuitively too, like<ins style="font-weight: bold; text-decoration: none;">: </ins>as long as F has no jumps, you can get nearby values of F by adding lots of or little bits of area.]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) - 0 = \int_a^x f</math> as we wanted.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) - 0 = \int_a^x f</math> as we wanted.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3403&oldid=prevIssaRice at 02:44, 8 September 20212021-09-08T02:44:31Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now what if F is some function such that F(a) is not 0? then we can define G(x) := F(x) - F(a), so that G(a)=0. Apply the argument/visualization above, to get <math>G(x)= \int_a^x G'</math>. But <math>G' = F'</math> so <math>F(x) - F(a) = \int_a^x F'</math>, and we are done.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now what if F is some function such that F(a) is not 0? then we can define G(x) := F(x) - F(a), so that G(a)=0. Apply the argument/visualization above, to get <math>G(x)= \int_a^x G'</math>. But <math>G' = F'</math> so <math>F(x) - F(a) = \int_a^x F'</math>, and we are done.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">the default way to picture a function is to draw it as a curve. in FTC1, this works well. you have a function (curve). you integrate it (area under curve). then you differentiate it (height of curve). you get back the original curve. happy ending.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">but now what if you have a function, then want to differentiate it, then take the integral of it? if you start with a curve, now you need to visualize the slope, and how the slope changes along the curve. that's hard. instead of doing the hard thing, you cheat. if you start with an area, then you can visualize the derivative as a curve (height). then you take the integral, so you get area.</ins></div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3402&oldid=prevIssaRice at 02:15, 8 September 20212021-09-08T02:15:45Z<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:15, 8 September 2021</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area. [PROOF?]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area. [PROOF? <ins style="font-weight: bold; text-decoration: none;">-- hmm, a rigorous proof of this ''might'' just require FTC2, which makes this visualization circular... but you can think of it intuitively too, like... as long as F has no jumps, you can get nearby values of F by adding lots of or little bits of area.</ins>]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) - 0 = \int_a^x f</math> as we wanted.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) - 0 = \int_a^x f</math> as we wanted.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now what if F is some function such that F(a) is not 0? then we can define G(x) := F(x) - F(a), so that G(a)=0. Apply the argument/visualization above, to get <math>G(x)= \int_a^x G'</math>. But <math>G' = F'</math> so <math>F(x) - F(a) = \int_a^x F'</math>, and we are done.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now what if F is some function such that F(a) is not 0? then we can define G(x) := F(x) - F(a), so that G(a)=0. Apply the argument/visualization above, to get <math>G(x)= \int_a^x G'</math>. But <math>G' = F'</math> so <math>F(x) - F(a) = \int_a^x F'</math>, and we are done.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3401&oldid=prevIssaRice at 02:10, 8 September 20212021-09-08T02:10:53Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>There's a typical picture of FTC1 that you see in places like Pugh's analysis book or 3Blue1Brown's video on FTC. This explanation makes sense, but I want point out a few different ways of thinking about the picture. One is, like the 3b1b video says, to look at the incremental area. You get <math>\Delta A(x) \approx f(x)\Delta x</math>. So then you divide by <math>\Delta x</math> and take the limit as <math>\Delta x \to 0</math> and get <math>A'(x) = \lim_{\Delta x \to 0} \frac{\Delta A(x)}{\Delta x} = f(x)</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>There's a typical picture of FTC1 that you see in places like Pugh's analysis book or 3Blue1Brown's video on FTC. This explanation makes sense, but I want point out a few different ways of thinking about the picture. One is, like the 3b1b video says, to look at the incremental area. You get <math>\Delta A(x) \approx f(x)\Delta x</math>. So then you divide by <math>\Delta x</math> and take the limit as <math>\Delta x \to 0</math> and get <math>A'(x) = \lim_{\Delta x \to 0} \frac{\Delta A(x)}{\Delta x} = f(x)</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Another way of looking at this that I saw in one of John Stillwell's books is that <math>\int f(x) \, dx</math> is the sum of infinitely many quantities <math>f(x)\, dx</math>. So the incremental thing you add is <math>f(x)\, dx</math>, i.e. <math>d \int f(x) \, dx = f(x)\, dx</math>. Now if you divide by <math>dx</math> you get <math>\frac{d \int f(x) \, dx}{dx} = f(x)</math>, which again is FTC1.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Another way of looking at this that I saw in one of John Stillwell's books <ins style="font-weight: bold; text-decoration: none;">[i think it was mathematics and its history, in the calculus chapter, in the section talking about leibniz] </ins>is that <math>\int f(x) \, dx</math> is the sum of infinitely many quantities <math>f(x)\, dx</math>. So the incremental thing you add is <math>f(x)\, dx</math>, i.e. <math>d \int f(x) \, dx = f(x)\, dx</math>. Now if you divide by <math>dx</math> you get <math>\frac{d \int f(x) \, dx}{dx} = f(x)</math>, which again is FTC1.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Finally, let's look at <math>A(x) = \int_a^x f(t)\,dt</math>. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, changes, as the width of the rectangle changes, is the height of that rectangle".</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Finally, let's look at <math>A(x) = \int_a^x f(t)\,dt</math>. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, changes, as the width of the rectangle changes, is the height of that rectangle".</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3400&oldid=prevIssaRice at 02:09, 8 September 20212021-09-08T02:09:28Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:09, 8 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area. [PROOF?]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area. [PROOF?]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) = \int_a^x f</math> as we wanted.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) <ins style="font-weight: bold; text-decoration: none;">- 0 </ins>= \int_a^x f</math> as we wanted<ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Now what if F is some function such that F(a) is not 0? then we can define G(x) := F(x) - F(a), so that G(a)=0. Apply the argument/visualization above, to get <math>G(x)= \int_a^x G'</math>. But <math>G' = F'</math> so <math>F(x) - F(a) = \int_a^x F'</math>, and we are done</ins>.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3399&oldid=prevIssaRice at 02:07, 8 September 20212021-09-08T02:07:37Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:07, 8 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
<td colspan="2" class="diff-lineno">Line 7:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area<ins style="font-weight: bold; text-decoration: none;">. [PROOF?]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Now, what is the height of this graph? why, by FTC1 it must be that the height is F', which is f. so <math>F(x) = \int_a^x f</math> as we wanted</ins>.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3398&oldid=prevIssaRice at 02:05, 8 September 20212021-09-08T02:05:06Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:05, 8 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math><ins style="font-weight: bold; text-decoration: none;">. So how do we visualize this? well the trick is, let's say you start with a function F. how the heck do you visualize it? well one way is to let F be a curve. then F' becomes the instantaneous slope. but it's hard to visualize that slope changing over time... so an alternative is, you let F be ''area'', and then f just becomes the ''height'' (i.e., the curve). that allows us to visualize ''both'' functions nicely, just like in FTC1. but now, if we fix some point a, then let the area from a to x be F(x), we have a problem. because now F(a) is an area of 0, so we can't represent an arbitrary function F. so the trick is... for the moment, let's make the problem easier on ourselves by requiring that F(a)=0. Then, as long as F is "smooth" enough, we can represent F as an area</ins>.</div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3397&oldid=prevIssaRice at 02:01, 8 September 20212021-09-08T02:01:29Z<p></p>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">So, let's be clear about what we are trying to show. If <math>F</math> is some nice function, and is an antiderivative of <math>f</math> (i.e., <math>F'=f</math>), then we have <math>\int_a^b f(x)\, dx = F(b) - F(a)</math>.</ins></div></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3396&oldid=prevIssaRice at 01:59, 8 September 20212021-09-08T01:59:32Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:59, 8 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5">Line 5:</td>
<td colspan="2" class="diff-lineno">Line 5:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Finally, let's look at <math>A(x) = \int_a^x f(t)\,dt</math>. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, changes, as the width of the rectangle changes, is the height of that rectangle".</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Finally, let's look at <math>A(x) = \int_a^x f(t)\,dt</math>. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, changes, as the width of the rectangle changes, is the height of that rectangle".</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2. <ins style="font-weight: bold; text-decoration: none;">It bothers me that people are so enthusiastic about showing you visualizations of FTC1, but then they ''don't even mention'' how to visualize FTC2. Like, they probably ''don't'' have a visualization, so then they are like "well let's just be quiet about it here and nobody will ask". Even 3b1b does this, and i am like >:( wtf. So anyway, I think FTC2 can be '''visualized using the same picture as FTC1''', but you just have to sort of use your brain in a different way? like you have to think about it in a different way, but it's the same picture!</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2.</div></td><td colspan="2" class="diff-side-added"></td></tr>
</table>IssaRicehttps://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Fundamental_theorem_of_calculus&diff=3395&oldid=prevIssaRice at 01:55, 8 September 20212021-09-08T01:55:22Z<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:55, 8 September 2021</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td>
<td colspan="2" class="diff-lineno">Line 3:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Another way of looking at this that I saw in one of John Stillwell's books is that <math>\int f(x) \, dx</math> is the sum of infinitely many quantities <math>f(x)\, dx</math>. So the incremental thing you add is <math>f(x)\, dx</math>, i.e. <math>d \int f(x) \, dx = f(x)\, dx</math>. Now if you divide by <math>dx</math> you get <math>\frac{d \int f(x) \, dx}{dx} = f(x)</math>, which again is FTC1.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Another way of looking at this that I saw in one of John Stillwell's books is that <math>\int f(x) \, dx</math> is the sum of infinitely many quantities <math>f(x)\, dx</math>. So the incremental thing you add is <math>f(x)\, dx</math>, i.e. <math>d \int f(x) \, dx = f(x)\, dx</math>. Now if you divide by <math>dx</math> you get <math>\frac{d \int f(x) \, dx}{dx} = f(x)</math>, which again is FTC1.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Finally, let's look at <math>A(x) = \int_a^x f(t)\,dt</math>. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, changes, is the height of that rectangle".</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Finally, let's look at <math>A(x) = \int_a^x f(t)\,dt</math>. What is the rate of change of this area function? As x changes, A(x) changes a bit. The rate of change is jut the height of the graph, since the area increases "one vertical line at a time". Or you can think of it as, "the rate at which area, approximated as a rectangle, <ins style="font-weight: bold; text-decoration: none;">changes, as the width of the rectangle </ins>changes, is the height of that rectangle".</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Now, on to FTC2.</div></td></tr>
</table>IssaRice