Problem 6 - Calculus

Question

Find the Taylor polynomials of degree \(n\) approximating the functions for \(x\) near \(0 .\) (Assume \(p\) is a constant. \()\) $$\ln (1+x), \quad n=5,7,9$$

Step-by-Step Solution

Answer

The Taylor polynomials are: - Degree 5: \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} \) - Degree 7: \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} \) - Degree 9: \( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^8}{8} + \frac{x^9}{9} \).

Step 1: Understanding Taylor Polynomials

The Taylor polynomial of a function \( f(x) \) at a point \( a \) is given by \( P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)(x-a)^2}{2!} + \ldots + \frac{f^{(n)}(a)(x-a)^n}{n!} \). For this exercise, we will approximate \( \ln(1+x) \) around \( x = 0 \).

Step 2: Find Derivatives of ln(1+x)

First, find the necessary derivatives of \( f(x) = \ln(1+x) \). Differentiate to get:- \( f(x) = \ln(1+x) \)- \( f'(x) = \frac{1}{1+x} \)- \( f''(x) = -\frac{1}{(1+x)^2} \)- \( f'''(x) = \frac{2}{(1+x)^3} \)- \( f''''(x) = -\frac{6}{(1+x)^4} \) and so on up to the 9th derivative.

Step 3: Evaluate Derivatives at x=0

Next, evaluate the derivatives at \( x = 0 \):- \( f(0) = \ln(1+0) = 0 \)- \( f'(0) = 1 \)- \( f''(0) = -1 \)- \( f'''(0) = 2 \)- \( f''''(0) = -6 \)- Continue this pattern to get values up to \( f^{(9)}(0) \), which will be alternately positive and negative factorials.

Step 4: Formulate the Taylor Polynomial for n=5

Using the evaluated derivatives, form the Taylor polynomial of degree 5:\[ P_5(x) = 0 + 1x - \frac{x^2}{2} + \frac{2x^3}{3!} - \frac{6x^4}{4!} + \frac{24x^5}{5!} \]Simplifying yields:\[ P_5(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} \]

Step 5: Formulate the Taylor Polynomial for n=7

Following the same procedure, the Taylor polynomial of degree 7 is:\[ P_7(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} \]

Step 6: Formulate the Taylor Polynomial for n=9

Finally, for degree 9, we extend the pattern:\[ P_9(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^8}{8} + \frac{x^9}{9} \]

Key Concepts

Maclaurin Series

The Maclaurin series is a special case of the Taylor series where the function is approximated at zero, i.e., when the point of expansion \( a \) is zero. This is particularly useful when dealing with functions like \( \ln(1+x) \), which are naturally expanded around zero for simplicity and convenience.
When expanding a function in a Maclaurin series, you use the formula:

• \( f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \ldots + \frac{f^{(n)}(0)x^n}{n!} \)

For the function \( \ln(1+x) \), its Maclaurin series expansion involves finding the derivatives at zero and then building the polynomial using these derivatives. As detailed, the series forms the basis for approximating logarithmic functions near zero by providing polynomial expressions for better understanding and computation ease.

• It helps solve complex problems by simplifying the function into a polynomial.
• The higher the degree of the series, the closer the approximation to the actual function.

Using the Maclaurin series allows us to estimate values of \( \ln(1+x) \) effectively, which can be crucial in calculations where precision and simplicity are needed.

Logarithmic Functions

Logarithmic functions include expressions like \( \ln(1+x) \), a common function studied within mathematics because of its natural properties. A logarithmic function is the inverse of an exponential function and is characterized by its ability to scale multiplicative relationships into additive ones.
In the context of calculus:

• Logarithmic functions are smooth and continuous.
• They grow slower than polynomial functions as \( x \to \infty \).

The derivative of a logarithmic function, such as \( \ln(1+x) \), leads directly to rational functions that are important in series approximations, such as the Taylor or Maclaurin series. Understanding the derivatives and behavior of logarithmic functions is crucial for creating accurate polynomial approximations.
These functions are extensively used in different fields such as:

• Physics, by describing growth processes like radioactive decay.
• Computer science, in algorithms that relate to time complexity.

Thus, knowing how to handle and approximate them via series expansion is a core mathematical skill.

Function Approximation

Function approximation is an essential tool in mathematics that involves representing a complex function with a simpler polynomial form. This method is especially useful when dealing with functions that are difficult to compute directly.
Using Taylor or Maclaurin polynomials, we approximate functions in the following ways:

• We develop truncated polynomial expansions that give us approximate values of the function within a specific range.
• The degree of the polynomial (e.g., \( n = 5, 7, 9 \)) influences how closely the polynomial matches the actual function.

In real-world applications:

• Such approximations allow for simpler calculations on calculators and computers.
• They enable numerical methods like integration and differentiation to proceed with less computational complexity.

The process begins by identifying the first few derivatives of the function, evaluating these at the point of interest (often zero for Maclaurin series), then assembling these into the polynomial. The method balances between accuracy and simplicity, emphasizing the utility of polynomials to approximate functions efficiently.