Why Are Slope and Area Opposite: The Fundamental Theorem of Calculus

[@danieldonnell3162]

They should actually just print this explanation on the cover of calc 1 textbooks... Very concise and clear. Thanks!

[@whatitmeans]

3Blue1Brown has a beautiful video about this topic, but your explanation is much more intuitive. Congrats.

[@hi-mv7eg]

This was a brillant explanation never really understood what slope had to do with calculus thanks for clearing it up.

[@themonrovian8441]

After doing an entire physics degree I never saw an explanation as clear as this for illustrating the fundamental theorem of calculus. Bravo sir 👏

[@SATMathReview1234]

Brilliant, amazing we live in an era where such profound things that took so much genius and effort to discover are readily accessible to anyone.

[@benbearse4783]

The way that let me understand integrals the best is that the anti derivative of a function is literally the area formula for under the graph. Like take y=x for example. The distance between any point is X and the height of any point is Y which equals X. This forms a triangle because it’s just a straight diagonal line. The formula for the area of a triangle is bh/2 so base (x) times height(x) divided by 2 =(x^2)/2 Which I thought was really cool. This continues for every other possible line The area for under a quadratic is 1/3(bh) where base is still x and height is x^2 (hence y=x^2) so (x^3)/3 is the area

[@lyricass7810]

Your intuition is so much better than 3blue and this video, I don't understand why you didn't get single like, thanks for commenting bro your comment made my day, but I still don't understand how adding infinitesimally small rectangles is equal to taking anti derivative of a integral function f(x) 🥲. Why finding anti derivative will do the work of adding infinitesimally small rectangles? And how ? If you have intuition for this please let me know bro 🥲.

[@benbearse4783]

@@lyricass7810 Thank you! I really appreciate what you said. Honestly, I’m just glad that my comment could help at least one person. When it comes to the logic behind why the anti derivative gives the area could be best explained saying that, the derivative of a function is found by dividing the function by a tiny change dx, while the area is found by multiplying it by tiny changes dx(which by multiplying tiny change in x by the formula for y getting the area for the rectangle under that little instance of the graph), ultimately undoing what the derivative did. Hope this helps! If not i can try and clarify for you.

[@lyricass7810]

@@benbearse4783 thanks for the reply bro, I would say I understood 50 percent 😂, can you clarify clearly please. How anti derivative will take care of adding infinitesimally small rectangles with different areas 🥲. Thanks in advance,

[@bruhifysbackup]

​@@lyricass7810And that is where fundamental theorem of calculus steps in.

[@mmgedi]

Wow, as the width of the rectangle tends to zero, the height of the rectangle tends to the slope of the original function. In other words, that infinitesimal sliver becomes the tangent line at that specific instant of time. Thank you so much!

[@Yue27s]

Real thanks bro, you can explain it in really simple term, i dont know is that simple, real thanks!

[@bpatricksullivan]

Happy to help! Thank you for viewing.

[@sadornsamdi3263]

I have bien looking for such a video for quite à long time thank you

[@bpatricksullivan]

Glad it was helpful!

[@bob-pq6ef]

Excellent thank you

[@binodtharu8348]

Thanks a lot for ur explanation i am very happy right now that i was able to prove this result because of you liked and subscribed👍

[@myonn0319]

This should be in every calculus textbook!

[@savinunimeth53]

wow what an explanation

[@bpatricksullivan]

Thanks, Savinu. I hope the video is useful.

[@bismajoyosumarto1237]

Whoaaa thanks! :D That was quite short and clear 👍

[@bpatricksullivan]

Thank you, Bisma!

[@umachakraborty8568]

I totally understand calculus.☺️☺️☺️thanks.

[@JavierBonillaC]

That is really very good.Every single rectangle has as a height the original function and as width dx. So if the slope is constant (say a horizontal line) the area will alyas be dx times 1. I seem to understand. thank you so much for this super interesting video.

[@bpatricksullivan]

I am glad you found it helpful!

[@henrytzuo8517]

Thank you !! Very Clear!!

[@razvanlazu3371]

great explanation !

[@bpatricksullivan]

Thanks! Glad you found it useful.

[@gravitystorm58]

Loved the video, great explanation! I have a question though; when you divide by dx and then take the limit, on the right side of the equation you’d have something of the form 0/0, does this matter?

[@bpatricksullivan]

That is one of the most significant results of calculus. Often we have ratios of infinitesimals which we can evaluate in the contexts of limits, and although both numerator and denominator approach zero, they approach zero at different rates, and as a result the ratio remains finite and non-zero.

[@odysseasv7734]

Very nice video it helped me a lot!

[@hoteny]

Thanks. Even tho I am free from calc-1 now, it was a good watch.

[@bpatricksullivan]

Thank you; glad you enjoyed it!

[@complexlogarithm3993]

Outstanding work!

[@EE-Spectrum]

Brilliant! Thanks so much.

[@bpatricksullivan]

Glad it was helpful!

[@BilalAhmed-on4kd]

I have a q about this, we must've added the limit as dx->0 before the last step So we have Lim dx->0[g(x+h)-g(x)]=f(x)dx So when we divide by dx we have { Lim dx->0[g(x+h)-g(x)] }/dx=f(x) So what we actually have now is that the numerator applies only on the numerator of the lhs,which is not exactly what the derivative of a function is

[@slowfern]

very helpfull

[@florentinosanchez3969]

Best video ever, thank you so mucgh

[@bungarmy7201]

BRILLIANT !

[@Yguy]

I don't understand why such an important explanation was not in my book. Nice explanation though, Keep it up!

[@bpatricksullivan]

Glad it was helpful!

[@savazeroa]

Dang i came up with this myself at school

[@byronkenoly]

brilliant!

[@roygreen8265]

I think that great explaining ; but ill like to know, why in the first place you can call the area under the curve a function g(x)?

[@_maestroking]

As x is increased or decreased area under the curve will also increase or decrease that is for any change in x there is change in area so it is dependent of x or we can say area is the function of x

[@dnuma5852]

nice video but theres more things you can do in calculus than just integrals and derivatives and they have a lot more applications than just finding tangent lines and areas

[@samueldeandrade8535]

The example was not good. Like, not good at all.