Problem 35 - Calculus Early Transcendentals

Question

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\ln x, a=e$$

Step-by-Step Solution

Answer

Question: Construct the nth-order Taylor polynomials for the function \(f(x) = \ln x\) at the point \(a = e\), and find the Taylor polynomials for \(n=0,1,\) and 2. Then, graph the Taylor polynomials and the original function. Answer: The Taylor polynomials for \(n = 0, 1,\) and 2 are as follows: - \(n=0\): \(P_0(x) = 1\) - \(n=1\): \(P_1(x) = 1 + \frac{1}{e}(x-e)\) - \(n=2\): \(P_2(x) = 1 + \frac{1}{e}(x-e) - \frac{1}{2e^2}(x-e)^2\) To graph these Taylor polynomials and the original function, plot the functions \(f(x) = \ln x\), \(P_0(x) = 1\), \(P_1(x) = 1 + \frac{1}{e}(x-e)\), and \(P_2(x) = 1 + \frac{1}{e}(x-e) - \frac{1}{2e^2}(x-e)^2\) together. You will observe that the Taylor polynomials provide better approximations of the original function near the point \(a = e\) as the order of the polynomial increases.

Step 1: Find derivatives of the function at the given point

First, we need to find the 0th, 1st, and 2nd-order derivatives of \(f(x) = \ln x\) and evaluate them at the point \(a = e\). - 0th derivative: \(f(x) = \ln x \Rightarrow f(a) = f(e) = \ln e = 1\) - 1st derivative: \(f'(x) = \frac{1}{x} \Rightarrow f'(a) = f'(e) = \frac{1}{e}\) - 2nd derivative: \(f''(x) = -\frac{1}{x^2} \Rightarrow f''(a) = f''(e) = -\frac{1}{e^2}\)

Step 2: Calculate the Taylor polynomials

Now that we have the derivatives evaluated at the point \(a = e\), we will use the formula for the Taylor Polynomial to create the polynomials of order 0, 1, and 2. - \(n=0\): $$P_0(x) = f(a) = 1$$ - \(n=1\): $$P_1(x) = f(a) + \frac{f'(a)}{1!}(x-a) = 1 + \frac{1}{e}(x-e)$$ - \(n=2\): $$P_2(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 = 1 + \frac{1}{e}(x-e) - \frac{1}{2e^2}(x-e)^2$$

Step 3: Graph the Taylor polynomials and the original function

To graph the Taylor polynomials and the original function, we will plot the following functions together: 1. \(f(x) = \ln x\) 2. \(P_0(x) = 1\) 3. \(P_1(x) = 1 + \frac{1}{e}(x-e)\) 4. \(P_2(x) = 1 + \frac{1}{e}(x-e) - \frac{1}{2e^2}(x-e)^2\) Using graphing software or graphing calculator, plot these four functions together. You should see that the Taylor polynomials provide better approximations of the original function near the point \(a = e\) as the order of the polynomial increases.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, representing the rate of change of a function at any given point. When we talk about derivatives in the context of Taylor Polynomials, we are specifically interested in evaluating these derivatives at a certain point, known as the center, to create an approximation of a function. For the function \( f(x) = \ln x \), and centered at \( a = e \), we found:

• The 0th derivative \( f(x) = \ln x \), evaluated at \( a = e \), gives \( f(e) = \ln e = 1 \).
• The 1st derivative \( f'(x) = \frac{1}{x} \), evaluated at \( a = e \), gives \( f'(e) = \frac{1}{e} \).
• The 2nd derivative \( f''(x) = -\frac{1}{x^2} \), evaluated at \( a = e \), gives \( f''(e) = -\frac{1}{e^2} \).

Derivatives allow us to construct Taylor Polynomials, as they provide the necessary coefficients that define how the polynomial approximates the function near a point.

Graphing Taylor Polynomials

Graphing Taylor Polynomials is a powerful way to visualize how these polynomials approximate a function. By graphing the original function \( f(x) = \ln x \) along with its Taylor polynomials, one can see how each polynomial gets closer to the curve of the original function near the center point, \( a = e \).

• The 0th-order Taylor Polynomial \( P_0(x) = 1 \) is simply a horizontal line at \( y = 1 \).
• The 1st-order Taylor Polynomial \( P_1(x) = 1 + \frac{1}{e}(x-e) \) forms a tangent line that touches the graph of \( \ln x \) at \( x = e \).
• The 2nd-order Taylor Polynomial \( P_2(x) = 1 + \frac{1}{e}(x-e) - \frac{1}{2e^2}(x-e)^2 \) curves to better match the shape of \( \ln x \) near \( x = e \).

By plotting these on graphing software or a calculator, it's evident that higher-order polynomials provide a closer fit to the original function near the center. This visual evidence supports the mathematical theory that Taylor Polynomials are useful for approximating functions.

Approximation of Functions

The main purpose of Taylor Polynomials is to approximate more complex functions using simpler polynomial expressions. This approximation is particularly accurate close to the center point; in this case, centered at \( a = e \) for the natural logarithm function \( f(x) = \ln x \).

• The 0th-order polynomial is simply the function's value at the center, providing a basic approximation.
• The 1st-order polynomial adds a linear term, giving a slope to the approximation that matches the original function's slope at the center.
• The 2nd-order polynomial includes a quadratic term, which allows the approximation to curve and thus better fit the function's behavior near the center.

These polynomials help mathematicians and engineers approximate functions to various degrees of accuracy, depending on the needed precision and computational resources. As the order of the polynomial increases, the approximation becomes more accurate over a wider range around the center.