(a) Find the Taylor polynomials up to degree 5 for f (x) = sin x centered at a = 0. Graph f and

(a) Find the Taylor polynomials up to degree 5 for f (x) =...

Key Concepts

Taylor Series

A Taylor series is an infinite sum of terms calculated from the values of a function’s derivatives at a single point. It provides a way to represent smooth functions as power series, where each term is determined by the derivatives evaluated at the center of expansion. This series is central to understanding how functions can be approximated by polynomials locally.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series expanded about the point 0. It simplifies calculations since many functions have derivatives that are easier to compute at zero. The concept is important for approximating functions near the origin and is widely used in many areas of mathematics and applied sciences.

Polynomial Approximation

Polynomial approximation involves approximating a function by a polynomial of finite degree. Taylor polynomials are specific examples that capture the behavior of a function near a certain point. The degree of the polynomial determines the accuracy of the approximation over a given interval, making this concept key in numerical methods and analysis.

Convergence of Taylor Polynomials

Convergence refers to how the Taylor polynomials get closer to the actual function as more terms are added. This involves concepts like the radius of convergence and error estimation. Understanding convergence is crucial to determine the limits of polynomial approximations, especially when the evaluation point moves away from the center of expansion, as well as for ensuring that the approximation reliably represents the function.

Transcript

00:01 In this problem we are asked to expand sign effects using taylor's polynomial up to degree of 5.

00:08 We know that taylor's expansion of a series is of the form like this and we can approximate a given function of effects around a point a using derivatives and actually higher order derivatives.

00:23 So since we are needing to expand it up to order of 5, we need f prime effect.

00:31 F, double prime of x, f triple prime of x, f4 of x, as well as f5 of x.

00:46 All right.

00:47 We know that f of x is sine of x, so the first derivative of that would be cosine of x.

00:52 Derrative of cosine would be negative sine of x.

00:55 Derrative of negative sine is negative cosine of x.

00:58 Therewith 2 of negative cosine is sine of x.

01:01 And it has a cyclic pattern, as you can see now, derivative of x.

01:05 Fifth there with it would again be cosine of x.

01:07 Now let's say we'll let this function around 0.

01:13 So 1a is 0.

01:18 We have f prime of 0 as 1.

01:22 F double prime of 0 is 0.

01:25 F triple prime of 0 is negative 1.

01:29 F4 of 0 is 0 since sine 0 is 0 and cosine 0 is 1.

01:34 So f5 of 0 5 derivative of f would also be 1.

01:42 All right.

01:43 Now we have pretty much everything that we need.

01:46 All we need to do is to take these values and then plug it in to this equation.

01:52 Then we can estimate, approximate, sine of x as sine of 0 plus f prime of 0 is 1.

02:01 So 1 times x minus 0 plus x plus f second derivative is 0, so that is 0, plus third derivative is negative 1, so that is negative 1 of x minus 0, cube divided by 3 factorial.

02:19 4th derivative is 0, so we have plus 0, plus 5th derivative is 1.

02:29 So we have 1 times x minus 0 to the 5 divide by 5 factorial.

02:34 We know that sign 0 is also 0, so we can write this one as x minus x cube over 3 factorial plus x to the 5 factorial.

02:50 From this, what we see is this...

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