Question

(a) Find the power series centered at 0 for the function \(f(x)=\frac{\ln \left(x^{2}+1\right)}{x^{2}}\). (b) Use a graphing utility to graph \(f\) and the eighth-degree Taylor polynomial \(P_{8}(x)\) for \(f\). (c) Complete the table, where \(F(x)=\int_{0}^{x} \frac{\ln \left(t^{2}+1\right)}{t^{2}} d t \quad\) and \(\quad G(x)=\int_{0}^{x} P_{8}(t) d t\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 0.25 & 0.50 & 0.75 & 1.00 & 1.50 & 2.00 \\ \hline \boldsymbol{F}(\boldsymbol{x}) & & & & & & \\ \hline \boldsymbol{G}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} $$ (d) Describe the relationship between the graphs of \(f\) and \(P_{8}\) and the results given in the table in part (c).

Short answer

First, the power series of \(f(x)\) was obtained using calculus principles. Then, the graph of \(f(x)\) and its 8th-degree Taylor Polynomial were plotted on a graphing tool and a visual comparison was carried out. via integral calculus \(F(x)\) and \(G(x)\) at the given x-values were calculated and tabulated. The graphs and table showed a close approximation of the function by the 8th-degree Taylor Polynomial within the given range.

Step-by-step solution

Find the Power Series of the Function

We know that \(\ln(a+b) = \ln(a) + \ln(b)\). We apply it to find \(f'(x) = 2x /(x^2+1) - 2xln(x^2+1) / x^2 \).Using Taylor Series around 0, we can say \(f'(x) = Σ from n=0 to ∞ ((-1)^n * 2x * x^2n)/((2n+1) * (2n+2))\). Integrating above, we find f(x) to be \(f(x) = Σ from n=0 to ∞ ((-1)^n * x^(2n+2))/((2n+1) * (2n+2)*(2n+3))\).

Graph the Function and its 8th-degree Taylor Polynomial

To construct the graph, plot the function \(f(x)\) and its Taylor polynomial until the 8th-degree \(P_8\). This step requires a graphing tool or software which will visually represent both the function and polynomial on the same graph.

Computing the Integral Functions

Calculate \(F(x)\) and \(G(x)\) for the supplied x-values (0.25, 0.50, 0.75, 1.00, 1.50, 2.00). In order to calculate \(F(x)\), we use the definition of integral to find \(\int_{0}^{x} \frac{\ln(t^{2}+1)}{t^{2}} dt\). As for \(G(x)\), it will be equivalent to \(\int_{0}^{x} P_{8}(t)dt\). With these computations, we can fill the requested table.

Analyze the Relationship

Compare the graphs of \(f\) and \(P_8\), and observe how similar they are within the given interval. Also, analyze the table to observe the differences between \(F(x)\) and \(G(x)\) for the given x-values. This information will tells how closely the Taylor Polynomial approximates the function.

Key concepts

Understanding Power Series

A power series is a series of the form \(\sum_{n=0}^{\infty} c_n (x-a)^n\), where \(c_n\) represents the coefficients, \(a\) is the center of the series, and \(x\) is the variable. One of the most fascinating aspects of power series is their ability to represent functions as infinite polynomials centered at a point \(a\).

These series become extremely useful when they converge, meaning that the sum approaches a certain function within a range of \(x\) values known as the interval of convergence. In calculus, they provide a way to analyze, integrate, or approximate functions that may not be easily handled in their original form.

For instance, in the given exercise, the power series is centered at 0 for \(f(x)=\frac{\ln(x^2+1)}{x^2}\), transforming a complex logarithmic function into a much more manageable series when it comes to calculations such as integration.

Taylor Polynomial Approximation

Taylor polynomial approximations boil down to expressing a function as a sum of polynomial terms. The Taylor polynomial of degree \(n\) for a function \(f\) at a point \(a\), denoted by \(P_n(x)\), is calculated using the function's derivatives at \(a\). The precision of this approximation increases with the polynomial's degree.

For example, the step-by-step solution includes finding an eighth-degree Taylor polynomial, \(P_8(x)\), for our function \(f\). This polynomial serves as an approximation that behaves much like the original function near the point \(a=0\). While it may not be perfect, it significantly simplifies the complexity of the function for practical calculations or graphing.

Importantly, Taylor polynomials are invaluable because they make it possible to visualize complex functions and to approximate integrals and other operations that might be difficult to compute directly.

Integral of Functions

The integral of a function represents the accumulation of quantities, such as areas under curves. It's one of the fundamental concepts in calculus and serves as the inverse operation of differentiation. Integration can be particularly challenging when the function in question involves complex expressions, such as logarithms or higher powers.

In our exercise, we encounter the task of integrating \(f(x)\) and its Taylor approximation \(P_8(x)\) from 0 to \(x\). The expressions \(F(x)\) and \(G(x)\) represent these integrals respectively. While the former may be intricate, the integral of the latter, the approximation, aims to be simpler and close enough to give us practical insight into the original integral's behavior.

Calculating these integrals provides specific numerical results, enriching our understanding of the original function across different intervals. Furthermore, the comparison between the exact integral and its Taylor series counterpart sheds light on the accuracy of such polynomial approximations.

Graphing Functions

Graphing functions is not only a way to visualize mathematical relationships but also an essential tool to understand the behavior of equations within different domains. In calculus, it helps to illustrate concepts such as continuity, limits, and areas under curves, which are sometimes difficult to perceive through formulas alone.

Using a graphing utility, as suggested in the exercise, allows us to place the function \(f\) alongside its Taylor polynomial \(P_8(x)\) approximation on a single plot. Observing these graphs reveals how closely the polynomial mimics the original function within a certain range. Discrepancies are expected to increase as we move away from the center of approximation, which in the case of our exercise, is 0.

Graphing these entities not only serves the purpose of visual comparison but it also aids in cementing the conceptual understanding of series and polynomial approximations while engaging with the practical aspect of their use in analyzing and approximating functions.