Question

Approximate the function value with the indicated Taylor polynomial and give approximate bounds on the error. Approximate In 1.5 with the Taylor polynomial of degree 3 centered at \(x=1\).

Short answer

The approximate value is 0.41667 with an error less than 0.002.

Step-by-step solution

Understand the Taylor Polynomial

To approximate a function using a Taylor polynomial, we use the formula: \[P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^n(a)}{n!}(x-a)^n,\] where \(a\) is the center of the polynomial. Here, we need to use the function \(f(x) = \ln x\) centered at \(x=1\).

Calculate necessary derivatives

Since we want a degree 3 polynomial, we need the first, second, and third derivatives of \( f(x) = \ln x \):- \( f'(x) = \frac{1}{x} \),- \( f''(x) = -\frac{1}{x^2} \),- \( f'''(x) = \frac{2}{x^3} \).

Evaluate derivatives at center

Now, evaluate the derivatives at \(a = 1\):- \( f(1) = \ln 1 = 0 \),- \( f'(1) = 1 \),- \( f''(1) = -1 \),- \( f'''(1) = 2 \).

Construct Taylor polynomial

Substitute these into the Taylor polynomial formula:\[P_3(x) = 0 + 1(x-1) - \frac{1}{2}(x-1)^2 + \frac{2}{6}(x-1)^3 = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3.\]

Evaluate polynomial at desired point

Plug \(x = 1.5\) into the polynomial:\[P_3(1.5) = (1.5-1) - \frac{1}{2}(1.5-1)^2 + \frac{1}{3}(1.5-1)^3 = 0.5 - \frac{1}{2}(0.5)^2 + \frac{1}{3}(0.5)^3.\]Simplify the expression:\[P_3(1.5) = 0.5 - 0.125 + \frac{1}{24.3} = 0.41667.\]

Estimate the error

The error for a Taylor polynomial is bounded by:\[|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}\]where \( M \) is the maximum value of \(|f^{(n+1)}(c)|\) for \(c\) in the interval between \(a\) and \(x\). Here, the 4th derivative is \(-\frac{6}{x^4}\). Evaluating near \(x=1.5\), \(|f^{(4)}(1.5)| \approx 0.177\).Calculate the error bound:\[|R_3| \leq \frac{0.177 \times 0.5^4}{4!} \approx 0.002.\]

Conclusion on approximation and error

The approximate value of \(\ln 1.5\) using a third-degree Taylor polynomial centered at \(x=1\) is about 0.41667, with an error less than 0.002.

Key concepts

Taylor Polynomial

A Taylor polynomial is a way to approximate a difficult-to-calculate function with a simpler polynomial. The primary idea is to express a function as a series expansion around a specific point, known as the center of the polynomial. To do this, we use the Taylor series formula:\[P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots + \frac{f^n(a)}{n!}(x-a)^n\]Here, \(n\) indicates the degree of the polynomial, and \(a\) is the point about which the series is centered. By choosing a point close to the value where you want to approximate the function, the Taylor polynomial can provide a very accurate estimate while being computationally easier to handle.

Error Estimation

Error estimation in Taylor polynomials gives us insight into how accurate our polynomial approximation is. The error is bounded by:\[|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}\]Where \(M\) is the maximum value of the \((n+1)^{th}\) derivative of the function within the interval considered. This formula helps us understand how much the Taylor polynomial deviates from the actual function value. In practical terms, it tells us how much 'trust' we can place in our approximation. Calculating the magnitude of this error gives confidence in how close our polynomial is to the real function.

Derivative Calculation

In order to construct a Taylor polynomial, calculating derivatives is crucial. These derivatives inform the coefficients of the polynomial. For the function \(f(x) = \ln x\), we calculate the following derivatives:- First derivative: \(f'(x) = \frac{1}{x}\)- Second derivative: \(f''(x) = -\frac{1}{x^2}\)- Third derivative: \(f'''(x) = \frac{2}{x^3}\)These derivatives are then evaluated at the center, \(a\). Calculating and substituting these into the polynomial provides each term in the series. Understanding and calculating these derivatives are essential for building accurate Taylor polynomials.

Logarithmic Function

The logarithmic function, denoted \(\ln x\), is a common function encountered in calculus and mathematical analysis. It's the inverse of the exponential function, and it grows at a decreasing rate as \(x\) increases. When working with Taylor series, we often expand logarithmic functions because they can be challenging to compute directly, especially at values not simplifying neatly.Using logarithmic properties, we adapt the function for easier approximation. In this exercise, expanding \(\ln x\) using a Taylor polynomial helps simplify the evaluation and develop error bounds around the point \(x = 1\). The natural logarithm's derivatives progressively capture finer changes in the function's behavior.

Polynomial Approximation

Polynomial approximation involves using a polynomial to represent a more complex function. This makes it possible to apply straightforward arithmetic operations to estimate values. In the given problem, a third-degree Taylor polynomial seeks to approximate \(\ln 1.5\) based on the simpler function's properties around \(x=1\).This technique is highly beneficial because:

• Polynomials are easier to differentiate and integrate.
• They provide manageable methods for estimating values we cannot compute analytically.
• Higher-degree approximations usually lead to more precise estimations, though computation may become more complex.

Polynomial approximation is foundational in numerical analysis, enabling solutions for functions that don't have simple closed forms.