Approximating sinx Let f(x)=sinx, and let pn and qn be...

Approximating $\sin x$ Let $f(x)=\sin x,$ and let $p_{n}$ and $q_{n}$ be nth-order Taylor polynomials for $f$ centered at 0 and $\pi,$ respectively. a. Find $p_{5}$ and $q_{5}$. b. Graph $f, p_{5},$ and $q_{5}$ on the interval $[-\pi, 2 \pi] .$ On what interval is $p_{5}$ a better approximation to $f$ than $q_{5} ?$ On what interval is $q_{5}$ a better approximation to $f$ than $p_{5} ?$ c. Complete the following table showing the errors in the approximations given by $p_{5}$ and $q_{5}$ at selected points. $$\begin{array}{|c|c|c|} \hline x & \left|\sin x-p_{5}(x)\right| & \left|\sin x-q_{5}(x)\right| \\ \hline \pi / 4 & & \\ \hline \pi / 2 & & \\ \hline 3 \pi / 4 & & \\ \hline 5 \pi / 4 & & \\ \hline 7 \pi / 4 & & \\ \hline \end{array}$$ d. At which points in the table is $p_{5}$ a better approximation to $f$ than $q_{5} ?$ At which points do $p_{5}$ and $q_{5}$ give equal approximations to $f ?$ Explain your observations.

Approximating $\sin x$ Let $f(x)=\sin x,$ and let $p_{n}$ and $q_{n}$ be nth-order Taylor polynomials for $f$ centered at 0 and $\pi,$ respectively.
a. Find $p_{5}$ and $q_{5}$.
b. Graph $f, p_{5},$ and $q_{5}$ on the interval $[-\pi, 2 \pi] .$ On what interval is $p_{5}$ a better approximation to $f$ than $q_{5} ?$ On what interval is $q_{5}$ a better approximation to $f$ than $p_{5} ?$
c. Complete the following table showing the errors in the approximations given by $p_{5}$ and $q_{5}$ at selected points.
$$\begin{array}{|c|c|c|}
\hline x & \left|\sin x-p_{5}(x)\right| & \left|\sin x-q_{5}(x)\right| \\
\hline \pi / 4 & & \\
\hline \pi / 2 & & \\
\hline 3 \pi / 4 & & \\
\hline 5 \pi / 4 & & \\
\hline 7 \pi / 4 & & \\
\hline
\end{array}$$
d. At which points in the table is $p_{5}$ a better approximation to $f$ than $q_{5} ?$ At which points do $p_{5}$ and $q_{5}$ give equal approximations to $f ?$ Explain your observations.

Key Concepts

Error Analysis

Error analysis in Taylor polynomial approximations involves quantifying the remainder or the difference between the actual function and its polynomial approximation. This error, often estimated by the Lagrange remainder formula, helps in assessing the quality and limitations of the approximation on various intervals and under different conditions.

Graphical Comparison of Approximations

A graphical comparison involves plotting both the original function and its Taylor polynomial approximations to visually inspect and determine where each approximation performs better. By comparing graphs, one can identify intervals where one polynomial offers a closer match to the function than another, illustrating the practical impact of the chosen center of expansion on the accuracy of the approximation.

Taylor Polynomial

A Taylor polynomial is a finite truncation of the Taylor series and serves as an approximation to the original function near the center of expansion. By using a polynomial where higher degree terms improve accuracy, one can obtain a practical estimation of the function's value, especially when calculating with elementary functions like sine or cosine.

Taylor Series

A Taylor series is an infinite sum representing a function as the sum of its derivatives at a single point, each multiplied by a corresponding power of the difference from that point divided by the factorial of the order. This series converges to the function within a certain interval, making it a powerful tool for approximating complex functions using simpler polynomial forms.

Center of Expansion

The center of expansion is the specific point around which the Taylor series (and its polynomial approximation) is computed. The choice of this center critically determines the region where the approximation is most accurate. When approximating functions, using a center closer to the evaluation point typically results in lower error, whereas a center far from the evaluation point may lead to significant deviation.

Transcript

00:01 In this question, we need to find the fifth order teta polynomial center at different points.

00:09 So let's first take the derivative for f at the beginning.

00:16 And we need to take the derivative up to fifth order.

00:20 Since it's a basic function, so it's not that difficult to do.

00:35 Okay, so this is, these are the derivatives of f.

00:41 Up to fifth order.

00:47 Now if we consider the center to be 0, so if 0 equals to 0, the first of the derivative, as 0 equals to 1, the second of the directive equals to 0, third order directive equals to minus 1, fourth of the derivative equals 0, and the last piece is 1.

01:13 And if we choose the center to be, pi the first of the derivative equals 0 this so the original function equals 0 the 1 the 1 second derivative 0 3rd order derivative 1 4th order derivative 0 5th order derivative 0 5th order derivative equals to minus 1 so according to this by the notation we denotes the taylor polynomial center that equals 0 by p5 which is so we only have three terms survived so the linear term x minus the cubic term 1 over 6 x cubed and the fifth of the term plus 1 over 120 x3 5 so all of this coefficient comes from the factorials and we denote the taylor polynomial centered pi by q5 again, only three terms survive, the linear term, cubic term.

02:30 The center should be pi.

02:33 So we have minus x minus pi.

02:37 And then we have a cubic term.

02:43 We have a fifth order term.

02:50 So these two are p5 and the q5.

02:52 We need to figure out.

02:56 Once we have this, we can graph if the original function and these two tailor polynomials on the same coordinate system.

03:07 So it looks like this.

03:10 Let's label some points...