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: Order 0: 0, Order 1: x, Order 2: x - x^2/2, Order 3: x - x^2/2 + x^3/3.

Step 1: Understand Taylor Polynomial

The Taylor polynomial of function \( f(x) \) around \( a \) is given by: \[ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n \] where \( f^n(a) \) is the \( n \)-th derivative of \( f \) at \( x = a \).

Step 2: Compute Derivatives of the Function

Given \( f(x) = \ln(1+x) \), we compute the first few derivatives:1. \( f(x) = \ln(1+x) \), so \( f(0) = \ln(1+0) = 0 \) 2. \( f'(x) = \frac{1}{1+x} \), so \( f'(0) = 1 \) 3. \( f''(x) = -\frac{1}{(1+x)^2} \), so \( f''(0) = -1 \) 4. \( f'''(x) = \frac{2}{(1+x)^3} \), so \( f'''(0) = 2 \).

Step 3: Construct the Taylor Polynomials

Using the Taylor polynomial formula, build each polynomial of orders 0, 1, 2, and 3:1. Order 0: \( P_0(x) = f(0) = 0 \)2. Order 1: \( P_1(x) = f(0) + f'(0)x = 0 + 1\cdot x = x \)3. Order 2: \( P_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 = x - \frac{x^2}{2} \)4. Order 3: \( P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 = x - \frac{x^2}{2} + \frac{x^3}{3} \).

Key Concepts

ln(1+x) function

The function \ln(1+x) is a natural logarithm function, where the natural logarithm is the logarithm to the base \( e \). The base \( e \) is an irrational and transcendental number which is approximately 2.71828. The function \ln(1+x) is defined only for \( x > -1 \) because the argument of a logarithm must be positive. One interesting characteristic of \ln(1+x) is that at \( x = 0 \), it evaluates to 0, making the computation of its derivatives at 0 especially useful for series expansions, such as the Taylor series.

• Defined for \( x > -1 \)
• \( \ln(1+0) = 0 \)
• Used in series expansions

Understanding this function helps in creating approximations of more complex functions using polynomials, which forms the basis of Taylor series. The \ln(1+x) function is commonly used as an introductory example in series expansion, which is pivotal in both mathematical analysis and practical applications in fields like engineering and physics.

derivatives

Derivatives represent the rate at which a function is changing at any given point, and they are fundamental in constructing Taylor polynomials. To find a Taylor polynomial around a point \( a \), you'll need to compute several derivatives of the given function.
For the function \( f(x) = \ln(1+x) \), here's how the derivatives calculate at \( x = 0 \):

• The first derivative \( f'(x) = \frac{1}{1+x} \), gives the slope of the tangent line, and evaluated at \( x=0 \) is 1.
• The second derivative \( f''(x) = -\frac{1}{(1+x)^2} \), representing the curvature or concavity, evaluates to -1 at \( x=0 \).
• The third derivative \( f'''(x) = \frac{2}{(1+x)^3} \), evaluated at \( x=0 \) is 2, and this process continues for higher-order derivatives.

Each derivative plays a crucial role in developing higher-order Taylor polynomials, improving the accuracy of the approximation over a wider interval. Understanding derivatives and their calculations is vital for using Taylor polynomials to solve complex analytical problems.

order of polynomial

The order of a polynomial in the context of Taylor series refers to the degree of the highest power of \( x \) in the polynomial. The order determines the accuracy of the polynomial's approximation to the function near the point \( a \).
The Taylor polynomials of different orders for \( f(x) = \ln(1+x) \) around \( a = 0 \) include:

• Order 0: Constant polynomial \( P_0(x) = 0 \).
• Order 1: Linear polynomial \( P_1(x) = x \).
• Order 2: Quadratic polynomial \( P_2(x) = x - \frac{x^2}{2} \).
• Order 3: Cubic polynomial \( P_3(x) = x - \frac{x^2}{2} + \frac{x^3}{3} \).

Higher-order polynomials capture more aspects of the function, resulting in better approximations. However, each increase in order requires computing additional derivatives, which can be complex.
Understanding polynomial order is important for accurately approximating functions and determining how many terms to include to meet a desired accuracy level.

Taylor series expansion

Taylor series expansion is a method to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This allows complex functions to be expressed as polynomials, which are easier to manipulate, integrate, and differentiate.
The general formula for a Taylor series is:
\[ f(x) \approx P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n \]
For the function \( \ln(1+x) \):

• At \( a = 0 \), the Taylor series expansion is a straightforward way to approximate \( \ln(1+x) \) for small values of \( x \).
• Each term adds more accuracy by accounting for higher-order behavior of the function at point \( a \).

It is important because it simplifies the complex behavior of functions into polynomials, which developers use in numerical analysis, physics, and engineering. Understanding this concept broadens the ability to estimate solutions to problems that might otherwise be too complex to solve analytically.