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The Taylor polynomial is an essential tool in approximating functions locally around a point. To construct a Taylor polynomial of degree 3 for a function like \( f(x) = x \ln x \) at point \( a = 1 \), you start by calculating the derivatives up to the third order. These derivatives capture the behavior of the function at the center \( a \). Each derivative contributes to a term in the polynomial:- The first derivative, \( f'(x) = \ln x + 1 \), forms the linear term.- The second derivative, \( f''(x) = \frac{1}{x} \), contributes to the quadratic term.- The third derivative, \( f'''(x) = -\frac{1}{x^2} \), leads to the cubic term. After calculating and evaluating them at \( a = 1 \), substitute into the Taylor polynomial formula:\[ T_3(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3 \]By plugging in the evaluated derivatives, you get:\[ T_3(x) = (x-1) + \frac{1}{2}(x-1)^2 - \frac{1}{6}(x-1)^3 \]This polynomial acts as a truncated version of the function, giving a snapshot of its behavior near \( a = 1 \).

Taylor's Inequality provides a way to estimate the error when using a Taylor polynomial. It tells us how close the Taylor polynomial is to the actual function over a specified interval. The inequality is expressed as:\[ |R_n(x)| \leq \frac{M}{(n+1)!} |x-a|^{n+1} \]Where \( M \) is the maximum value of the \((n+1)\)th derivative of the function in the interval of interest.To apply this, consider the fourth derivative, \( f^{(4)}(x) = \frac{2}{x^3} \). By evaluating its absolute maximum over the interval \([0.5, 1.5]\), we determine \( M \). In this case, \( M = 16 \) at \( x = 0.5 \). Plugging into Taylor's inequality gives:\[ |R_3(x)| \leq \frac{16}{24}|x-1|^4 \]This establishes an upper bound for the error within the interval, ensuring that the Taylor polynomial accurately approximates \( f(x) = x \ln x \) around \( x = 1 \).

Determining how accurate a Taylor polynomial is in approximating a function relies heavily on error estimation. Once we have calculated the error bound using Taylor's Inequality,we know how much the Taylor polynomial might deviate from the actual function within the interval.The error term, \( |R_3(x)| = \left| x \ln x - T_3(x) \right| \), measures the difference between the actual function and its Taylor polynomial approximation. By assessing the error,we can see how well the polynomial fits the function over \([0.5, 1.5]\). This approximation must be quick and efficient, especially in practical applications like physics or engineering,where exact values might not be easily computable.Thus, the error estimation validates the use of Taylor polynomials in approximating complex functions with a quantifiable confidence level.

Graphing the error term helps visually verify how accurate the Taylor polynomial is. To do this, plot \( |R_3(x)| = \left| x \ln x - T_3(x) \right| \) over the interval \([0.5, 1.5]\). This involves comparing the real function \( x \ln x \) and its approximation \( T_3(x) \).By graphing:

• The curve of \( x \ln x \),
• The Taylor polynomial \( T_3(x) \),
• The absolute error \( |R_3(x)| \),

Visualizing these functions showcases how close the polynomial is to the actual function. This comparison not only helps in understanding the behavior of the approximating polynomial but also provides practical insights on intervals where the approximation is most accurate.Such visual analysis can support theoretical findings, making it easier for students and professionals to grasp the utility of Taylor polynomials in real-world scenarios.