Best center point Suppose you wish to approximate cos(π/ 12) using Taylor polynomials. Is the ap

Best center point Suppose you wish to approximate cos(π/ 12)...

Best center point Suppose you wish to approximate $\cos (\pi / 12)$ using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at $\pi / 6 ?$ Use a calculator for numerical experiments and check for consistency with Theorem $11.2 .$ Does the answer depend on the order of the polynomial?

Key Concepts

Taylor Series

The Taylor series is a method for representing a differentiable function as an infinite sum of terms calculated from its derivatives at a single point. It lets us approximate functions with polynomials, where higher-order polynomials typically yield closer approximations to the original function within a region near the expansion point.

Center of Expansion

The center of expansion is the point at which all derivatives of the function are evaluated to generate the Taylor series. Placing the center closer to the target evaluation point generally leads to a more accurate approximation because the error terms, which depend on the distance from the center, are smaller.

Remainder (Error) Term

The remainder term, often given by the Lagrange form in Taylor's theorem, quantifies the error between the function's actual value and its Taylor polynomial approximation. The theorem provides a bound on this error based on the next higher derivative, the order of the polynomial, and the distance from the center of expansion.

Error Analysis and Polynomial Degree

Error analysis involves comparing the size of the remainder term for different centers of expansion and polynomial orders. While increasing the polynomial degree generally improves accuracy, the proximity of the expansion center to the evaluation point is crucial. The error bound, as described in the relevant theorem, demonstrates that even a lower-degree polynomial can provide a better approximation if the expansion center is closer to the point of interest.

Transcript

00:01 For this problem, we define, we choose the function f to be e to dx, and we consider two different taylor polynomial.

00:12 They centered a different point.

00:16 But first, we just take the derivative of f, and we see that all of its derivatives are the same.

00:28 All of them are e to dx.

00:31 So consider the center, a equals to 0.

00:41 So all of its derivatives at 0 equals to 1.

00:50 So p and x equals to the summation, n from k from 0 to n, the k -thal derivative equals to 1.

01:02 So 1 over k factorial times x to the k.

01:06 So this is the first theta polymer we are looking for.

01:11 And for the second one, we can see the center a equals to loan of 2.

01:20 So in this case, all of the derivative equals to 2.

01:33 So the corresponding taylor polynomial we denote by qx equals to the summation and k from 0 to n, 1 over k factorial times x minus long of 2 to the k.

01:50 Okay, so we can compare the performance for this two by calculating of soda error.

02:02 Since we are looking for the estimation for f of 0 .35, so at x equals to 0 .35, we compute f minus p n and f minus qn.

02:26 So here is the result.

02:29 So the first row in this table will be the other end.

02:35 So we compare several different n.

02:38 For example, a equals to 1, and equals 2, n equals to 3, any equals to 4, and a equal to 5.

02:48 Now next we consider the first taylor polym, which is centered at a equals to 0.

02:56 So if a equals to 0, when n equals to 0, one, the absolute error will be 6 .9 times 1020 minus 2.

03:04 You can do this by calculator...

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