Problem 4 - Thomas Calculus

Question

Find the Taylor polynomials of orders \(0,1,2,\) and 3 generated by \(f\) at \(a .\) \(f(x)=\ln (1+x), \quad a=0\)

Step-by-Step Solution

Answer

The Taylor polynomials are 0, x, x - \frac{x^2}{2}, and x - \frac{x^2}{2} + \frac{x^3}{3}.

Step 1: Understand the Taylor Polynomial Formula

The Taylor polynomial of degree \(n\) for a function \(f(x)\) centered at \(a\) is given by \( T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n \).

Step 2: Calculate First Few Derivatives of the Function

Calculate the derivatives of \(f(x) = \ln(1+x)\) with respect to \(x\) at \(x = 0\):- \(f(x) = \ln(1+x)\) so \(f(0) = \ln(1) = 0\)- \(f'(x) = \frac{1}{1+x}\) so \(f'(0) = 1\)- \(f''(x) = -\frac{1}{(1+x)^2}\) so \(f''(0) = -1\)- \(f'''(x) = \frac{2}{(1+x)^3}\) so \(f'''(0) = 2\)

Step 3: Construct the 0th Order Taylor Polynomial

The 0th order Taylor polynomial only includes the constant term: \( T_0(x) = f(0) = 0 \).

Step 4: Construct the 1st Order Taylor Polynomial

Add the first derivative term to the constant term: \( T_1(x) = f(0) + f'(0)(x - 0) = 0 + 1 \cdot x = x \).

Step 5: Construct the 2nd Order Taylor Polynomial

Include up to the second derivative term: \( T_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 = 0 + x + \frac{-1}{2}x^2 = x - \frac{x^2}{2} \).

Step 6: Construct the 3rd Order Taylor Polynomial

Include up to the third derivative term: \( T_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 = 0 + x - \frac{x^2}{2} + \frac{2}{6}x^3 = x - \frac{x^2}{2} + \frac{x^3}{3} \).

Key Concepts

Taylor Series

Taylor series help us approximate complex functions with polynomials. This means representing a function as an infinite sum of terms. These terms are derived from the function's derivatives at a single point.
For a function \( f(x) \), its Taylor series centered at \( a \) is:

• \( T(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \)

The beauty of Taylor series lies in its ability to replicate the behavior of a function near a specific point by using derivatives. Each derivative gives insight into the function's behavior. When adding more terms, the polynomial starts looking more like the actual function. Remember, a Taylor polynomial of order \( n \) uses terms up to the \( n \)-th derivative.

Derivatives

Derivatives represent the rate at which a function is changing at any given point. They are essential in constructing Taylor series because they provide information about the function's slope and curvature.
For \( f(x) = \ln(1+x) \), here are the first few derivatives:

• First derivative: \( f'(x) = \frac{1}{1+x} \); this represents the slope of the tangent line to \( f \) at any \( x \).
• Second derivative: \( f''(x) = -\frac{1}{(1+x)^2} \); reflects the curvature or concavity.
• Third derivative: \( f'''(x) = \frac{2}{(1+x)^3} \); provides more intricate details of the curve's shape.

These derivatives help construct the polynomial by determining how each term behaves as we move away from the center point. Calculating them with respect to \( x \) gives us values that are crucial for approximating \( f(x) \).

Polynomial Approximation

Polynomial approximation aims to create a simpler version of a function that is easier to work with. Through the Taylor polynomial, we approximate \( ln(1+x) \) using simple polynomial terms instead of the complicated original function.
Starting with the Taylor polynomial of order 0, we only have the constant term. As we increase the degree of the polynomial to 1, 2, and 3, we include more derivatives, adding more terms. Each term added corresponds to the next derivative, offering increased accuracy. For an example:

• 0th order: \( T_0(x) = 0 \).
• 1st order: \( T_1(x) = x \).
• 2nd order: \( T_2(x) = x - \frac{x^2}{2} \).
• 3rd order: \( T_3(x) = x - \frac{x^2}{2} + \frac{x^3}{3} \).

These approximations are particularly useful in mathematics and physics, where finding exact values might be complex or unnecessary.

Natural Logarithm

The natural logarithm function, denoted as \( ln(x) \), is a special logarithm with base \( e \), where \( e \) is an irrational constant approximately equal to 2.718. This function is useful because it provides a way to easily deal with exponential growth and decay processes.
When considering \( f(x) = \ln(1+x) \), it's particularly simple near \( x = 0 \). The natural logarithm's properties make Taylor series an effective tool for understanding and working with \( ln \).
The choice of using \( ln(1+x) \) for Taylor series is not arbitrary. It is due to the function's behavior around the center point \( a = 0 \), making it straightforward to evaluate and useful for approximations in mathematical problems.