Question

The first three Taylor polynomials for \(f(x)=\sqrt{1+x}\) centered at 0 are \(p_{0}=1, p_{1}=1+\frac{x}{2},\) and \(p_{2}=1+\frac{x}{2}-\frac{x^{2}}{8} .\) Find three approximations to \(\sqrt{1.1}\).

Short answer

The first three approximations to √1.1 using Taylor polynomials are: 1. p0(0.1) = 1 2. p1(0.1) = 1.05 3. p2(0.1) = 1.48 (rounded to two decimal places)

Step-by-step solution

Substitute x = 0.1 into the first Taylor polynomial p0

Using the first Taylor polynomial \(p_0 = 1\), we get \(f(0.1) \approx p_0(0.1) = 1.\)

Substitute x = 0.1 into the second Taylor polynomial p1

Using the second Taylor polynomial \(p_1 = 1 + \frac{x}{2}\), we substitute \(x=0.1\) and find that \(f(0.1) \approx p_1(0.1) = 1 + \frac{0.1}{2} = 1.05.\)

Substitute x = 0.1 into the third Taylor polynomial p2

Lastly, using the third Taylor polynomial \(p_2=1+\frac{x}{2}-\frac{x^2}{8}\), we substitute \(x=0.1\) and find that \(f(0.1) \approx p_2(0.1) = 1 + \frac{0.1}{2} - \frac{(0.1)^2}{8} = 1.05 - \frac{1}{80} = 1. \bar{48}\). Thus, we obtained three approximations for \(\sqrt{1.1}\) using the first three Taylor polynomials: 1. \(p_0(0.1) = 1\) 2. \(p_1(0.1) = 1.05\) 3. \(p_2(0.1) = 1. \bar{48}\)

Key concepts

Taylor Series

A Taylor series is a powerful tool in mathematical analysis to approximate complex functions with polynomial expressions, which are much simpler to calculate. The series is named after the British mathematician Brook Taylor. When we say that the series is 'centered at 0', this means that the approximation is built around the point x=0, known as the expansion point.

The formula for the Taylor series of a function at a point a is:\[\begin{equation} f(a) + f^{\text{'}}(a)(x-a) + \frac{f^{\text{''}}(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + \cdots\end{equation}\]Each term in the series involves higher derivatives of the function at the point a, which gives an increasingly accurate approximation as more terms are included.

Function Approximation

Function approximation is the process of finding simpler functions that closely mimic the behavior of more complex functions over a certain domain. Taylor polynomials are a common method used for function approximation. By using derivatives of a function at a specific point, we develop a series that can approximate the function with a desired level of accuracy. This is particularly useful in situations where the original function is too complex to deal with directly.

The more terms from the Taylor series we use, the closer we can get to the actual function's value, within a certain range around our expansion point. Polynomial approximations make it practical to perform calculations that would otherwise be too difficult to compute, especially for functions that do not have simple algebraic forms.

Square Root Approximation

Approximating the square root of a number is a common application of Taylor polynomials. In the example problem, we want to approximate \(\sqrt{1+x}\), which is not straightforward to calculate for every x. The Taylor series gives us a way to approximate this function using a polynomial.

For small values of x, even the first or second polynomial can give us a good approximation. As we increase the number of terms from the Taylor series, the approximation becomes more precise. This is because the Taylor series captures the curvature and other characteristics of the square root function near the point of approximation.

Polynomial Approximation

Polynomial approximation is the representation of functions as polynomials to simplify calculation and analysis. The Taylor polynomials derived from Taylor series are examples of such approximations. Polynomial functions are preferred for approximation because their simple form - a sum of powers of x, makes them easy to compute, differentiate, and integrate.

In the exercise, the Taylor polynomials of degrees 0, 1, and 2 provide us with three different polynomial approximations for \(\sqrt{1+x}\) at \(x=0.1\). As we go from \(p_0\) to \(p_2\), we're incorporating more information about the behavior of the square root function, which enhances the approximation accuracy. The trade-off here is between computational simplicity and accuracy, which we balance depending on the needs of the problem we're solving.