Question

Approximieren Sie die Funktion \(f(x)=\frac{x}{\ln x}\) durch das TAYLOR-Polynom 2 . Grades in den Nähe von \(x_{0}=2\) und schätzen Sie die Approximationsgüte für \(x \in\left[2, \frac{11}{5}\right] \mathrm{ab}\).

Short answer

The Taylor polynomial is \( P_2(x) = \frac{2}{\ln 2} + \frac{\ln 2 - 1}{(\ln 2)^2}(x - 2) + \text{additional terms resulting from } f''(2) \). The approximation quality depends on the error, influenced by the third derivative near \( x_0=2 \).

Step-by-step solution

Determine Derivatives

To find the Taylor polynomial of the second degree for the function \( f(x) = \frac{x}{\ln x} \), we begin by computing the derivatives. First, find the first derivative \( f'(x) \) using the quotient rule: \( f'(x) = \frac{(\ln x) - \frac{x}{x}}{(\ln x)^2} = \frac{\ln x - 1}{(\ln x)^2} \). Next, calculate the second derivative \( f''(x) \) by also applying the quotient rule: \( f''(x) = \frac{d}{dx}\left(\frac{\ln x - 1}{(\ln x)^2}\right) \), which results in an expression that must be simplified.

Evaluate Derivatives at \( x_0 = 2 \)

Substitute \( x = 2 \) into the expressions for \( f(x) \), \( f'(x) \), and \( f''(x) \). We find that \( f(2) = \frac{2}{\ln 2} \), \( f'(2) = \frac{\ln 2 - 1}{(\ln 2)^2} \), and a simplified form of \( f''(2) \) must be evaluated. This is essential for constructing the Taylor polynomial.

Construct Taylor Polynomial

The second degree Taylor polynomial is given by \( P_2(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2}(x-x_0)^2 \). Using the computed derivatives: \( P_2(x) = \frac{2}{\ln 2} + \frac{\ln 2 - 1}{(\ln 2)^2}(x - 2) + \frac{f''(2)}{2}(x-2)^2 \). Simplify and rewrite this polynomial to approximate \( f(x) \).

Estimate the Approximation Quality

Evaluate the quality of the approximation over the interval \( [2, \frac{11}{5}] \). This involves estimating the error term of the Taylor polynomial, which is \( R_2(x) = \frac{f'''(c)}{6}(x-x_0)^3 \), for some \( c \) in \( [2, \frac{11}{5}] \). Since \( f'''(x) \) would be complicated to find exactly, consider the magnitude of the third derivative near \( x_0=2 \) to comment on the error.

Key concepts

Quotient Rule

The Quotient Rule is a fundamental tool in calculus used to find the derivative of a function that is a ratio of two differentiable functions. When dealing with a function such as \( f(x) = \frac{x}{\ln x} \), the rule helps us compute the derivative efficiently. The formula for the Quotient Rule is as follows: if \( g(x) = \frac{u(x)}{v(x)} \), then the derivative \( g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). This allows us to handle functions that are divided by each other with ease.

Important steps involve:

• Differentiating the numerator and denominator separately.
• Applying the rule by plugging into the formula.
• Simplifying the result to find the derivative of the function.

This technique is particularly useful when approximating functions with Taylor polynomials where derivatives up to a certain order are required.

Derivative Evaluation

Derivative evaluation is an essential step when constructing Taylor polynomials. Here, we work on derivatives of a function and substitute specific values to prepare for polynomial construction. For the function \( f(x) = \frac{x}{\ln x} \), evaluating at \( x_0 = 2 \) involves substituting \( x = 2 \) into the derivatives calculated using the quotient rule.

Here's how to approach it:

• Calculate the original function at \( x_0 \).
• Substitute \( x_0 \) into the first derivative to find \( f'(x_0) \).
• Substitute \( x_0 \) into the second derivative for \( f''(x_0) \).

This provides the needed coefficients for the Taylor polynomial. By evaluating these derivatives at the base point \( x_0 \), we facilitate accurate construction of the polynomial approximation.

Approximation Error

When using Taylor polynomials to approximate functions around a point, understanding the approximation error is crucial. The approximation error tells us how accurate our polynomial is in representing the function over a given interval. Generally, this is achieved by calculating the remainder term of the Taylor series.

For our function \( f(x) = \frac{x}{\ln x} \), the error is expressed as \( R_2(x) = \frac{f'''(c)}{6}(x-x_0)^3 \), where \( c \) is some point within the interval of approximation \([2, \frac{11}{5}]\). Evaluating the error involves:

• Estimating the third derivative of the function near \( x_0 \).
• Calculating the magnitude of the error based on the interval.

Understanding the error helps us determine the reliability of our polynomial and informs any necessary adjustments needed for high precision in approximations.

Mathematical Approximation

Mathematical approximation involves substituting complex functions with simpler ones to analyze or compute easily. Taylor polynomials are powerful tools in this context. They approximate functions around a specific point by equating derivatives of the function to those of the approximating polynomial.

The concept of approximation using Taylor polynomials requires:

• Understanding the function and derivatives at a known point (in our case, \( x_0 = 2 \)).
• Constructing the polynomial using derivatives up to the desired degree.
• Simplifying the polynomial to achieve close similarity to the original function.

This strategy allows us to work with complex functions in a simplified form, providing insights into behavior and making calculations more manageable, especially within a useful approximation range.