Question

Approximating \(\ln x\) Let \(f(x)=\ln x,\) and let \(p_{n}\) and \(q_{n}\) be the nth-order Taylor polynomials for \(f\) centered at 1 and \(e\), respectively. a. Find \(p_{3}\) and \(q_{3}\). b. Graph \(f, p_{3},\) and \(q_{3}\) on the interval (0,4]. c. Complete the following table showing the errors in the approximations given by \(p_{3}\) and \(q_{3}\) at selected points. $$\begin{array}{|c|c|c|} \hline x & \left|\ln x-p_{3}(x)\right| & \left|\ln x-q_{3}(x)\right| \\ \hline 0.5 & & \\ \hline 1.0 & & \\ \hline 1.5 & & \\ \hline 2 & & \\ \hline 2.5 & & \\ \hline 3 & & \\ \hline 3.5 & & \\ \hline \end{array}$$ d. At which points in the table is \(p_{3}\) a better approximation to \(f\) than \(q_{3}\) ? Explain your observations.

Short answer

Based on the graph and the error table, analyze which Taylor polynomial provides a better approximation at different points in the interval. At points closer to \(x = 1\), the Taylor polynomial \(p_3(x)\) will likely provide a better approximation, as it is centered at 1. At points closer to \(x = e\), the Taylor polynomial \(q_3(x)\) will likely provide a better approximation, as it is centered at \(e \approx 2.718\). By comparing the errors for each point, you can confirm this behavior. As the point of interest gets closer to one of the center points, the corresponding Taylor polynomial will typically provide a lower error, giving a better approximation to the original function \(f(x) = \ln x\).

Step-by-step solution

Compute the derivatives of \(f(x)\)

We need the first three derivatives of \(f(x)\) to compute \(p_{3}\) and \(q_{3}\). The first derivative is \(f'(x) = \frac{1}{x}\), and likewise, to find other derivatives: \(f''(x) = -\frac{1}{x^2}\) \(f'''(x) = \frac{2}{x^3}\)

Compute the Taylor polynomials

To find the nth-order Taylor polynomial of a function, we must compute the values of the function and its derivatives at the center, starting from order 0 and moving up to the given order. For this exercise, we are interested in order 3 polynomials. Evaluating these derivatives at \(x = 1\) and \(x = e\): 1. Centered at 1: \(f(1) = 0\), \(f'(1) = 1\), \(f''(1) = -1\), \(f'''(1) = 2\) \(p_3(x) = f(1) + f'(1)(x - 1) + f''(1)\frac{(x - 1)^2}{2!} + f'''(1)\frac{(x - 1)^3}{3!}\) \(p_3(x) = x - 1 - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{6}\) 2. Centered at \(e\): \(f(e) = 1\), \(f'(e) = \frac{1}{e}\), \(f''(e) = -\frac{1}{e^2}\), \(f'''(e) = \frac{2}{e^3}\) \(q_3(x) = f(e) + f'(e)(x - e) + f''(e)\frac{(x - e)^2}{2!} + f'''(e)\frac{(x - e)^3}{3!}\) \(q_3(x) = 1 + \frac{x - e}{e} - \frac{(x - e)^2}{2e^2} + \frac{2(x - e)^3}{6e^3}\) #b. Graph \(f, p_{3},\) and \(q_{3}\) on the interval (0,4]# To graph these functions on the interval (0, 4], you can input them into a graphing calculator or software such as Desmos or GeoGebra. 1. For \(f(x) = \ln x\), plot the curve on the interval (0, 4] 2. For \(p_3(x)\), plot the curve on the interval (0, 4] 3. For \(q_3(x)\), plot the curve on the interval (0, 4] By comparing these graphs, you will see how well \(p_3(x)\) and \(q_3(x)\) approximate \(f(x)\) on this interval. #c. Complete the table showing the errors in the approximations# We will compute the absolute errors at each \(x\) value in the table. The error at point \(x\) is defined as \(|\ln x - p_{3}(x)|\) for \(p_{3}\) and \(|\ln x - q_{3}(x)|\) for \(q_{3}\).

Compute the errors for each point

For each selected point, compute the errors using the expressions derived above. 1. x = 0.5: The errors are \(|\ln 0.5 - p_{3}(0.5)|\) and \(|\ln 0.5 - q_{3}(0.5)|\) 2. x = 1: The errors are \(|\ln 1 - p_{3}(1)|\) and \(|\ln 1 - q_{3}(1)|\) 3. x = 1.5: The errors are \(|\ln 1.5 - p_{3}(1.5)|\) and \(|\ln 1.5 - q_{3}(1.5)|\) 4. x = 2: The errors are \(|\ln 2 - p_{3}(2)|\) and \(|\ln 2 - q_{3}(2)|\) 5. x = 2.5: The errors are \(|\ln 2.5 - p_{3}(2.5)|\) and \(|\ln 2.5 - q_{3}(2.5)|\) 6. x = 3: The errors are \(|\ln 3 - p_{3}(3)|\) and \(|\ln 3 - q_{3}(3)|\) 7. x = 3.5: The errors are \(|\ln 3.5 - p_{3}(3.5)|\) and \(|\ln 3.5 - q_{3}(3.5)|\) Insert these errors in the table. #d. Determine which Taylor polynomial provides a better approximation#

Analyze the error table

Compare the errors for each point and determine which polynomial provides a better approximation. In your explanation, consider the center points and the relative proximity of the points of interest to those centers. The polynomial centered at the point closer to the point of interest will typically give better approximation.

Key concepts

Taylor Series Approximation

Taylor series provide a powerful tool for approximating complex functions using polynomials. The basic premise of a Taylor series approximation is to expand a function into an infinite sum of terms, calculated from the derivatives of the function at a single point, known as the center. This approach transforms functions that may be difficult to compute or integrate into a more manageable polynomial form.

For function understanding, consider that each term of a Taylor series incorporates a higher derivative of the function, scaled by a factor that diminishes with each additional term. The more terms we use, the closer the polynomial will approximate the function, at least near the center point. The n-th order Taylor polynomial is simply the Taylor series truncated after the n terms, giving us a finite approximation of the function.

For example, to approximate \(\ln x\), its Taylor polynomial at x = 1 or e would involve computing the function's derivatives at these points, then constructing a polynomial that adds these derivatives, adjusted by the appropriate factorial denominator and the \(x - c\) term, where c is the central point.

Natural Logarithm ln(x)

The natural logarithm, denoted as \(\ln x\), is a fundamental mathematical function with extensive applications in various scientific fields, including physics, engineering, and economics. It is the inverse of the exponential function \(e^x\), meaning that if \(y = \ln x\), then \(e^y = x\). The natural logarithm has a unique property of converting multiplicative relationships into additive ones, facilitating the handling of problems involving growth rates or time decay.

The Taylor polynomials for \(\ln x\) can be particularly insightful when learning about the behavior of \(\ln x\) near different points. Since the logarithmic function isn't defined for \(x \leq 0\), the approximation will be more accurate within a particular range where \(x > 0\), and especially close to the center of the Taylor polynomial. Understanding the properties of \(\ln x\), including its rate of change (derivative), is critical when working with the function's Taylor approximations.

Error Analysis

Error analysis in the context of Taylor polynomials is the study of how well the polynomial approximates the function it represents. The 'error' is the difference between the actual value of the function and the estimated value provided by the polynomial. Calculating the absolute error using \(\left| f(x) - p_n(x) \right|\) helps us understand and quantify the difference at particular points.

When conducting error analysis, it's important to assess where the approximation is most accurate and where it deviates significantly. Typically, a Taylor polynomial centered at point a will provide a better approximation close to a. As we move further away from the center, the error may increase. By comparing Taylor polynomials centered at different points, as with \(p_3\) and \(q_3\) from the exercise, we can determine which polynomial provides a better approximation over a particular interval or at specific points within that interval.

The table in the given exercise is a practical tool for error analysis. By filling in the absolute errors for each specified point, students can visualize the performance of each approximation. This hands-on approach enhances comprehension of the approximation's limitations and the consequences of choosing one center over another for the Taylor polynomial.