Problem 7 - Calculus: Early Transcendentals

Question

Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = \ln x, \) \( a = 1 \)

Step-by-Step Solution

Answer

The Taylor polynomial is \( T_3(x) = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 \).

Step 1: Function and Derivatives

We need to find the first few derivatives of the function \( f(x) = \ln x \) at the point \( a = 1 \). First derivative: \( f'(x) = \frac{1}{x} \)Second derivative: \( f''(x) = -\frac{1}{x^2} \)Third derivative: \( f'''(x) = \frac{2}{x^3} \)

Step 2: Evaluate Derivatives at a=1

Now, evaluate the derivatives at \( x = 1 \).\( f(1) = \ln 1 = 0 \)\( f'(1) = \frac{1}{1} = 1 \)\( f''(1) = -\frac{1}{1^2} = -1 \)\( f'''(1) = \frac{2}{1^3} = 2 \)

Step 3: Write the Taylor Polynomial

Use the formula for a Taylor polynomial:\[ T_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 \]Substituting the values, we get:\[ T_3(x) = 0 + 1(x-1) + \frac{-1}{2}(x-1)^2 + \frac{2}{6}(x-1)^3 \]Simplifying this expression:\[ T_3(x) = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 \]

Step 4: Graph the Function and Polynomial

Plot both \( f(x) = \ln x \) and \( T_3(x) = (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 \) on the same graph using graphing software or a calculator. The polynomial should approximate the function near \( x = 1 \).

Key Concepts

Derivatives

Derivatives are vital in calculus, representing the rate at which a function changes at any given point. When dealing with the function \( f(x) = \ln x \), derivatives can help us explore how the curve behaves at different points. They depict how rapidly the function's outputs alter as the input changes.

• The first derivative, \( f'(x) = \frac{1}{x} \), shows the slope of the function at any point \( x \).
• The second derivative, \( f''(x) = -\frac{1}{x^2} \), aids in understanding the concavity or convexity of \( f \).
• The third derivative, \( f'''(x) = \frac{2}{x^3} \), offers insights into the function's curvature.

When evaluating these derivatives at \( x = 1 \), we gather more information about the behavior of the function around this point, crucial for constructing a Taylor polynomial.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a fundamental function in mathematics, particularly in calculus.

• It is the inverse of the exponential function, specifically the \( e^x \) function.


• As \( x \) grows larger, \( \ln x \) increases slowly, reflecting the logarithmic growth pattern.
• At \( x = 1 \), \( \ln 1 = 0 \). This fact makes it a suitable center point (\( a \)) for Taylor polynomials since the function's behavior is smooth and predictable around this point.

Understanding the properties of the natural logarithm is crucial when approximating its value using polynomials.

Polynomial Approximation

Polynomial approximation is a technique used to estimate complex functions using polynomials, which are simpler and often easier to analyze.

• By constructing a polynomial, we can approximate a function like \( \ln x \) near a specific point \( a \), providing valuable insights into the function's behavior.
• The Taylor polynomial, such as \( T_3(x) \), helps us simplify and approximate more challenging calculations, especially when functions are difficult to evaluate directly.

These approximations are most accurate near the center of expansion and may differ further away from it. The closer \( x \) is to \( a \), the more exact the approximation is likely to be.

Taylor Series

A Taylor series is an infinite sum of terms calculated from the values of a function's derivatives at a particular point. It is a robust tool for representing functions as polynomials.

• The series allows us to express functions like \( \ln x \) as a sum of polynomial terms about \( a = 1 \).
• A Taylor polynomial, such as \( T_3(x) \), is a truncated version of the Taylor series. It includes a finite number of terms for practical purposes.

The formula for generating Taylor polynomials uses derivatives to construct increasingly more accurate estimates of the function. Though a true Taylor series would be infinite, paring down the series makes it applicable to real-world problems.

Graphing Functions

Graphing functions is a crucial step in visualizing the behavior of mathematical expressions, including functions like \( \ln x \) and their polynomial approximations.

• By plotting functions and their Taylor polynomials on the same graph, we can visually assess how closely the polynomial matches the original function near the center point.
• The behavior of \( \ln x \) and its approximation, \( T_3(x) \), shows how well the approximation works around \( x = 1 \).

Observing both plots reveals where the approximation is most accurate and where it diverges, offering insights into the function's properties and the effectiveness of the Taylor polynomial.