Do you eventually plan to go in "calculus proper" ? Yes, finding tangents to curves is the fundamental problem of diff. calculus, and the videos are extremely cool and instructive, but what is interesting to see is you going in series, sequences, convergence ,limits/infinite (or lack thereof), differentiability, the integral, the rules and theorems, proofs of taylor's, derivation of basic transcendental functions. I.E all the places where you have a non-standard view.
@redpictureLA The problem is actually not that the approximation is closer or not. For any real world application in engineering or science you actually need to know how far off you are, in other words have a very clear measure of your error. (This is doable)
@redpictureLA Look at the remainder formulas for taylor polynomials, those are the tools to determine error bounds (you get a bound, not the exact error , should be obvious why if you think for example the expansion may represent an irrational) for an Nth order taylor aproximation. Taylor expansion is arguably the most important formula in applied math. I think in MF81 we will be given an intro to the basic ideas of bounding errors (just guessing from the tittle)
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omicron8251 1 week ago in playlist MathFoundations
Do you eventually plan to go in "calculus proper" ? Yes, finding tangents to curves is the fundamental problem of diff. calculus, and the videos are extremely cool and instructive, but what is interesting to see is you going in series, sequences, convergence ,limits/infinite (or lack thereof), differentiability, the integral, the rules and theorems, proofs of taylor's, derivation of basic transcendental functions. I.E all the places where you have a non-standard view.
DanPartelly 1 month ago
If the Taylor expansion were performed on a cubic BiPolyNumber would the approximation be closer?
redpictureLA 1 month ago
@redpictureLA The problem is actually not that the approximation is closer or not. For any real world application in engineering or science you actually need to know how far off you are, in other words have a very clear measure of your error. (This is doable)
DanPartelly 1 month ago
@DanPartelly Ah. So in the given example in this video, my approximation would be 33.44, and I can find out that my error is .002?
redpictureLA 1 month ago
@redpictureLA Look at the remainder formulas for taylor polynomials, those are the tools to determine error bounds (you get a bound, not the exact error , should be obvious why if you think for example the expansion may represent an irrational) for an Nth order taylor aproximation. Taylor expansion is arguably the most important formula in applied math. I think in MF81 we will be given an intro to the basic ideas of bounding errors (just guessing from the tittle)
DanPartelly 1 month ago in playlist MathFoundations
At 30:00, What about implicit differentiation to avoid +/- and square root derivatives? That seems a little closer to your method.
teavea10 1 month ago
@teavea10 That's right, we are doing an algebraic form of implicit differentiation.
njwildberger 1 month ago