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!
Great question. This is a complex topic. "open outflow boundary conditions", "zero gradient boundary conditions", or "absorbing boundary conditions" are all terms you can use to learn about ways people have approached this problem. They all have flaws. It is a very desirable boundary condition in many real world systems, but it presents significant difficulties in implementation.
Thank you so much! I am glad the algorithm brought you here as well! I have a lot more similar content in my pipeline over the next few weeks. Please let me know any topics you'd like to see covered within computational physics, data science, data visualization, Python programming, or adjacent areas.
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
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 🥲.
@@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.
@@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,
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?
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.
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
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.
Thank you so much, you really made this intuitive for me. However, is this also the way the integration rules are derived? With the Riemann sum? Just like differentiation rules can be derived with ( f(x+h) - f(x) ) / h as h approaches 0?
