>Derivative is the slope of the tangent lineWhat the fuck does that even mean and how does it relate to solving real world problems?
lines easy. non lines hard. tangent lines describe local behavior of non lines so that non lines easy again.
>>16537644Based
>>165376361. Look up the definition od "slope".2. Look up the definition of " tangent".3. Look up the heat equation or something.
>>16537636In actuality the tangent line at a point is defined by the derivative of the function at that point. It's just that the tangent line is a concept that can be understood intuitively without the derivative and serves as a good motivation for constructing the derivative.It's good for visualizing rates of change. If we graph a linear change in a quantity, the rate of change is visualized as the steepness of the line. This is fine because the rate of change is constant across the function, since it is linear.Of course, when the rate of change varies, how do you visualize it? Of course you can look at the relative steepness of the curve, but if you graph what the quantity would be if the rate of change at a given point remained constant (the tangent line) you can visualize the rate of change at that point directly.There are other utilities as well. Since on a small interval of the given point, the tangent line approximates the function itself, you can use it for small, local approximations. It is also use to find roots in a wide variety of cases with Newton's method using the same principle (the tangent line as a crude model of the graphs behavior).
>>16537636A chord is a line between two points on a function. A tangent line is a limit of a chord when the two points are brought close to each other.>In actuality the tangent line at a point is defined by the derivative of the function at that pointYou can define a tangent line purely geometrically as I did above without any explicit reference to derivatives. All you need is for the space to be a complete metric space (metric for the notion of distance and completeness for the limiting procedure). In fact, derivatives on generic manifolds are defined via this chord limiting procedure, not the other way around as you’re suggesting.
>>16538577meant to reply to >>16538572 as well
>>16537636look at a visual lmfao its just a line through a function
>>16538605There’s an infinite family of lines crossing a single point, retard. Only one of them is the tangent line.
>>16538634There are infinite points 0 distance away from any point. From your axiom and my axiom, there are infinitely many parallel tangent lines that would share the same functional description, if while crossing different points.
>>16538662>There are infinite points 0 distance away from any pointThere is only one. It is the point itself. This is one of the axioms of a metric btw. Formally d(x,y) = 0 if and only if x=y. Read an undergrad book on linear algebra before ever posting again.
>>16538674NTA, but that's true in 1D. Once you have a scalar function of 2 (or more) variables then you have an infinite number of tangent lines, whose bases produce a tangent plane.
>>16538677>Once you have a scalar function of 2 (or more) variables then you have an infinite number of tangent linesThat's now what he is claiming. He is saying that there are infinitely many points 0 distance away from a given point. That's now what distance is. You can have this occur on pseudo-Riemannian manifolds, in which case the set of all such points forms an algebraic variety in that manifold. The most well-known example is the lightcone in special relativity. But I am 100% sure this faggot doesn't know any of this. He's just saying retarded shit because he never got beyond "infinitesimals" in his engineering degree.As for your note, yes, obviously this generalizes to tangent spaces on points of a manifold. These tangent spaces, however, are unions of tangent lines of the set of all geodesics that pass through that point, so the construction is still formally valid.
>>16537644genius
>>16538682Points have no extension. There are definitely infinite points zero distance from eachother. I do like how you backpedal and backpedal and backpedal to justify your pseud meltdown.
>>16538728Please go ahead an define what you mean by "extension". Because I don't know what that rigorously means.>There are definitely infinite points zero distance from eachother.So you have a notion of distance that is different from the standard notion. Great. Please list your axioms of the distance function then.
>>16538743Your points take up "distance?"
>>16538749No, a distance is a function whose arguments are points. Basic shit, anon.
>>16538682>That's now what he is claiming. He is saying that there are infinitely many points 0 distance away from a given point. That's now what distance is.That's fair. Also, you don't need to go so crazy with non-Riemannian spaces, you can just have any non-metric space and you'll have (potentially countably infinite) points of distance zero away. A "trivial" example is SE2 (or really any mixed-polar space such that your distance is clearly defined in some sub-coordinate pair).
>>16538761I specifically mentioned pseudo-Riemannian manifolds for a reason. The existence of a metric is replaced with a weaker statement that there exists a non-degenerate bilinear form from the manifold to the reals. Every manifold in general relativity is such a manifold because the bilinear form is the proper time between events in spacetime. Such manifolds do admit a notion of two points having zero distance between them and the algebraic variety of these points is an orbit of the action of the local Lie group on the neighborhood of a point. In relativity, this is the fancy-schmancy way of saying that things that move at the speed of light will always remain moving at the speed of light and things that move below the speed of light can never move at the speed of light.
>>16538634That’s not what I said retard, do you even know what a derivative is?
>>16537636Slope is like a geometric rate of change so the slope of the line tangent is the geometric equivalent of the instantaneous rate of change.
>>16537654>>16538697Take a high school level calculus class retards
>>16539903i calculated your mom last night
>>16539792You clearly don’t because it’s not “just a line”. It’s a very special kind of line.
>>16537636>Filtered by 3 nested prepositional phrasesI'm always surprised by how many people actually get stuck at this stage. At least have the self awareness and shame to hide your idiocy and figure it out for your self. I promise it's not that hard. Please stop showing everyone your ass.
>>16540019elaborate, I want to hear this shit
>>16540568I already did. Work on your reading comprehension.
>>16540581Retarded “deflection” (which would be a compliment)
