The right-hand side, because this is just Our polynomial at c, at least is going to be equal Us the fact that at least p of c, the approximation at c, That's p of x isĪpproximation, but then we could try to go for havingĭerivative matched. Have- this would just be a horizontal line Going to be a constant, it should at leastįunction equals at c. It- our polynomial could just be- if it's just Our first approximation is that our polynomialĪt c should be equal to- or actually, even let me better So this is our x-axis- when x is equal to c. Want to approximate this function when x. ![]() What I want to do now isĮxpand it a little bit, generalize it a little bit, andįocus on the Taylor expansion at x equals anything. Maclaurin series or the Taylor series at x is equal to 0. But all of that wasįocused on approximating the function around The function even a little bit longer than that. ![]() Third-degree polynomial, maybe something that hugs Second-degree polynomial, you can get something that If you have aįirst-degree polynomial, you can at least get the You can approximate it with a horizontal line that Zero-degree polynomial, which is just a constant, How we can approximate aįunction around x is equal to 0 using a polynomial. Videos, we learned how we can approximateĪn arbitrary function, but a function that isĭifferentiable and twice and thrice differentiableĪnd all of the rest. And not just at a single point, but at all points? Then how could one analyze and use the operations, objects, definitions, properties, and patterns associated with this infinite polynomial representation of Sine to conclude that it indeed is equivalent to the Sine function. All you have is a polynomial that is claimed to approximate the Sine function for any input. ![]() What if you did not know how the Sine Taylor Series was constructed? You did not know things like it is made up of terms that allow the derivative to work out, etc. Perhaps there is another way to ask the question. ![]() The other components have a of pattern increasing polynomial degrees, as well as, the n'th term being divided by the n'th factorial to make the derivatives work out, but what role is this pattern playing in realizing the Sine pattern? Also, why is it that the more you add, the better the approximation to Sine? I do see this reflected in the Taylor series expansion by the repetitive derivatives in the terms, but seeing how Sine's pattern is enforced from the rest of the components of the terms is not clear. How does combining the derivatives with the other components of the terms make the Sine Taylor Series work? How can one interpret a process that takes the individual results of derivatives at a single point, attach them to their appropriate polynomial and then add them together? This is especially confusing knowing that Sine clearly has a repetitive nature to it. How is it that the equivalence of the Taylor Series and Sine derivatives at a single point enables us to take the results of the individual results of the Taylor Series version, and place them collectively into a summation of terms that correctly maps all of the inputs and outputs for Sine? I have been struggling to find a good explanation as to why or how this step in the process works. These terms are added together and the more we add, the better the approximation to Sine, not just at that single point - but for all the points that one can input for Sine. The process then (at a high level) uses the derivatives at this single point as one of several components that make up the terms in the Taylor series (other components include increasing polynomial degrees that purposefully map to the derivatives of the chosen expansion point). I get that the series starts with identifying a point from which to expand the series and having the derivatives of the Sine function (in this case) and its Taylor series expansion match at this point. My understanding is that a Taylor Series expansion can actually be equivalent to the Sine function ( I am aware that not all Taylor expansions equal the Function in question). I have yet to connect all of the parts of the Taylor Series into a sensible story when it comes to the Taylor Series for functions like Sine. Also, a written expression of where I am are lost with this topic is not trivial to me. Second It should be noted that I am writing this because I am confused - so I realize that my lack of understanding may cause me to mischaracterize a some things. Hi, first let me say thank you in advance for reviewing my questions.
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |