Follow the procedure in the text to show that the n th-order...

Key Concepts

Taylor Polynomial

A Taylor polynomial is a polynomial approximation of a function around a specific point, often denoted by a. It is constructed so that the polynomial and the function have the same value and a number of derivatives at that point, thereby giving a local approximation of the function that improves with a higher-order polynomial.

Derivative Matching

Derivative matching is the process of ensuring that the Taylor polynomial’s derivatives up to a given order coincide with the corresponding derivatives of the function at the point of expansion. This guarantees that the polynomial not only approximates the function's value at the point but also its local behavior through its derivative values.

Coefficient Calculation

The coefficients of the Taylor polynomial are determined by using the values of the function’s derivatives at the point of expansion. Specifically, for a function that is infinitely differentiable at a point a, the coefficient c_k for the kth term is given by the kth derivative of the function at a divided by k factorial, i.e., c_k = f^(k)(a) / k!.

Power Series Representation

A power series is an infinite series in the form of a sum of terms involving powers of (x - a). The Taylor polynomial can be viewed as the truncated version of this series, retaining only a finite number of terms, which provides an approximation to the function near the point a.

Transcript

00:01 Some function of effects we do a taylor polynomial expansion taylor polynomial expansion around a around some point a around a and this taylor polynomial expansion is of degree n degree n any or any let's call this expansion p sub n effects for any k that is less well or equal than zero and the bigger oracle than zero and smaller than n for any such k the kth derivative of the taylor polynomial expansion evaluated at array is equal to the kth derivative of the function evaluated at a so so well that is true because if you have this this piece of n the k the k b of the form the value of the function at 0 plus the value of the first derivative no at not at a plus the value of the first derivative of the function at a times x minus a plus the second value this value of the second directive of the function at a over 2 factorial times x minus a squared and and so one and up to degree n so the nth derivative of the function evaluated at a divided by n factorial times x minus a to the power of n so if we differentiate well this is this is not a this is just the taylor polynal expansion of the degree of the n and if we differentiate one time that function or we're just going to get that get rid of that and then the derivative of this is just once we will get f prime at a and then by this we have now the two comes down so two divided by two factorial that is one time times this constant times x minus a and then the remaining powers are going to be are going to have powers of x minus a so that you can write the rest of the terms times x minus a because those are derivatives of degree higher than two differentiating degree higher than two so if we evaluate this piece prime sub n a well we will get this f prime at a and then these terms since x is equal to a a minus a that is going to be zero and also these terms here because they have x minus a so so it does agree for one negative and now if we differentiate this polynomial are i times also let's say that we differentiate k times this polynomial will be well the only term that survives is the the one that has x to the power n minus n times to the power k and bigger values because if we differentiate this one k times all the rest what is outside is just a constant is this when you differentiate it's going to be this and higher order terms so x minus a times k and then that times another polynomial and then that times another polynomial let's call that polynomial h of x so that the case degree of this would be case the directive of this by the linearity so the directive of this will give us a k factorial and then that's it so k factorial go like k factorial so that these cancels and then when we differentiate this one well the derivative of this would be just that h of x but this h of x is in terms of as powers of x minus a and then well all the derivative in this term the k theory would be that times km factorial plus by the power rule by the product rule no to the k and then x minus a to the k and then the k the derivative of h so well this this polynomial is in terms of x minus a so that when we evaluate at at a we'll have we're going to have this constant and then this term is zero and then this is also zero so that for k smaller than n, the derivative of this, of the taylor polynomial of the bn.

07:18 The k's derivative evaluator at a day agrees with the derivative of the function.

07:25 The kth derivative of the function evaluator at a.

07:30 So we will have this...

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