I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half.
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what does the can look like with the minimized surface area with its original volume. How is it gonna be different from the original shape of the unoptimized can?
This is a particular case of the intermediate value theorem, called Bolzano's theorem. The intermediate value theorem says that for any real k, if f(a)<k<f(b), there is c=>f(c)= k.
Finally understood , before i was wondering why is it implict different like just shuffle the terms around u will reach it & it turns out u can do it but its very complicated to solve organise and would have bad accuracy & impossible in some cases. And the [f(x)] was simplest and best explaination
I dont quite understand, i understand in the sense of the chain rule. And i can physically solve them but i dont get mentally. If your original function is implicit and you substitute y for f(x), well thats not possible right? Because implicit means it cannot be expressed as a function without loss of detail. So how does this work in general, in this case i could assume its because a circle is symmetric.
Can't get why we have to use the theorem to prove. If it's possible to change to polar isn't it better than getting confused with all that inequalities? In most examples I've seen u get either 0 meaning limit exists or some function with θ and because it varies u say the limit does not exist. Is that enough to prove not continuity? I'm too confused
If the z parameter is dropped we have the area over a line in the x-y plane under a surface f(x,y) which should be greater than the area between the surface defined and the integral of the differential of arc length of the line from (0,0,0) to (1,2,3). If f(x,y) is the plane z=2 and x=t, y=t, (t goes from 0 to 1) we get the rectangle ( made of two triangles) with area 2√2. But if the z parameter =2t and x=t and y=t, then the area between the surface f(x,y) =2 and the parametrized line would seem to be the triangle of area√2 or one half of the total rectangular area. Formally correct but we seem to be in an extra dimension.
This approach takes a shortcut by making z1 = 0, would have been nice to see the more general solution where z1 != 0 and substituting z^2 = x^2 + y^2 is not as straightforward as the video makes it seem
