91Ó°ÊÓ

When we work with Taylor polynomials, the goal is to approximate a function with a polynomial close to its behavior near a specific point. In this problem, we are approximating the function \( f(x) = \sqrt{1+x^2} \) at \( x_0 = 0 \) using a Taylor polynomial of degree 2. This polynomial is a simple form that uses the function's value and its first few derivatives evaluated at \( x_0 \) to mimic the function's shape.
The accuracy of this approximation depends greatly on how far \( x \) is from \( x_0 \) and the behavior of higher-order derivatives. For the interval \([0, \frac{1}{5}]\), a second-degree polynomial is suitable, especially since the third derivative and subsequent derivatives are relatively small, indicating that their influence on the polynomial's accuracy is negligible within this range.

• The degree of the polynomial gives a balance between simplicity and accuracy.
• The approximation generally gets better as the interval around \( x_0 \) gets smaller.
• Checking higher-order derivatives helps in estimating how 'flat' the function is, which influences approximation.

A key part of forming a Taylor polynomial involves calculating derivatives. The function \( f(x) = \sqrt{1+x^2} \) requires derivatives to form its Taylor approximation. Each derivative gives insight into how the function behaves locally.
We start with the first derivative, \( f'(x) = \frac{x}{\sqrt{1+x^2}} \), which tells us about the slope of \( f \). The second derivative, \( f''(x) = \frac{1}{(1+x^2)^{3/2}} \), provides information about the curvature.

• The first derivative at \( x_0 = 0 \) is \( 0 \), meaning the slope is flat at this point.
• The second derivative at \( x_0 = 0 \) is \( 1 \), showing positive curvature, indicating the function is concave up at \( x_0 \).

This curvature information is crucial in shaping the Taylor polynomial, as it affects the coefficient of the \( x^2 \) term.

The Taylor series centered at \( x = 0 \) is called the Maclaurin series. For \( f(x) = \sqrt{1+x^2} \), the Maclaurin series gives us a simplified polynomial that represents the function's behavior near zero.
In this case, we derive the second-degree Maclaurin polynomial, which is
\[ T_2(x) = 1 + \frac{1}{2} x^2. \]
This polynomial utilizes the first and second derivatives:

• \( f(0) = 1 \) gives the constant term.
• \( f'(0) = 0 \) results in no linear \( x \) term.
• \( f''(0) = 1 \) contributes to the \( \frac{1}{2} x^2 \) term.

The Maclaurin series is particularly useful for computing approximate values of the function \( f \) over short intervals around \( x = 0 \), providing an easier computational method than working with non-polynomial functions.

Even though a Taylor polynomial can approximate a function well, it is essential to know the possible error, especially when using the polynomial in place of the function over a range. The remainder term, \( R_2(x) \), quantifies this error for a second-degree Taylor polynomial:
\[ R_2(x) = \frac{f'''(\xi)}{6} x^3, \]
where \( \xi \) is a point between \( 0 \) and \( x \). Calculating \( f'''(x) = -\frac{3x}{(1+x^2)^{5/2}} \), we determine the possible error.

• For small values of \( x \), such as within \([0, \frac{1}{5}]\), this error is minimal because \(|f'''(x)|\) remains small.
• The evaluation of \( f'''(0) = 0 \) confirms that higher curvature impacts are nearly negligible in this region.

This understanding ensures confidence in the Taylor polynomial's use and provides a tangible estimate of its reliability.