Problem 6 - Calculus for Biology and Medicine

Question

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function. $$ f(x)=\frac{1}{1+x}, n=4 $$

Step-by-Step Solution

Answer

The degree 4 Taylor polynomial for \( \frac{1}{1+x} \) at \( x=0 \) is \( 1 - x + x^2 - x^3 + x^4 \).

Step 1: Recall the definition of Taylor Series

The Taylor series of a function \( f(x) \) about \( x = a \) is given by \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n \). We are asked to find the Taylor polynomial of degree 4 centered at \( x = 0 \).

Step 2: Determine the function and its derivatives

The given function is \( f(x) = \frac{1}{1+x} \). First, compute the derivatives of \( f(x) \): - \( f(x) = (1+x)^{-1} \)- \( f'(x) = -(1+x)^{-2} \)- \( f''(x) = 2(1+x)^{-3} \)- \( f'''(x) = -6(1+x)^{-4} \)- \( f^{(4)}(x) = 24(1+x)^{-5} \)

Step 3: Evaluate derivatives at x = 0

Calculate each derivative at \( x = 0 \):- \( f(0) = 1 \)- \( f'(0) = -1 \)- \( f''(0) = 2 \)- \( f'''(0) = -6 \)- \( f^{(4)}(0) = 24 \)

Step 4: Construct the Taylor polynomial

Using the derivatives evaluated at \( x = 0 \), plug them into the Taylor polynomial formula:- \( P_4(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4 \)- \( P_4(x) = 1 - 1 \cdot x + \frac{2}{2} x^2 - \frac{6}{6} x^3 + \frac{24}{24} x^4 \)- Simplifying gives: \( P_4(x) = 1 - x + x^2 - x^3 + x^4 \)

Key Concepts

Taylor series

The Taylor series is a powerful tool used in mathematics to approximate complex functions using simpler polynomial functions. Essentially, when we have a function that is difficult to work with directly, we can represent it as an infinite sum of terms calculated from the values of its derivatives at a single point. This is especially useful for functions that are not algebraic or involve complex operations.

When creating a Taylor series, we choose a point around which to expand, typically called the center or expansion point. In our exercise, this point is at zero, often denoted by the term "Maclaurin series" when the expansion point is zero. The general formula for a Taylor series about a point \(a\) is:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n \)
  

Here, each term involves a derivative of the function evaluated at the point \(a\), multiplied by a factor of \( (x-a)^n \), and divided by \( n! \) (n factorial). The accuracy of this approximation increases as more terms are included.

derivatives

Understanding derivatives is crucial when working with Taylor series. Derivatives represent how a function changes as its input changes. In simple terms, a derivative tells us the rate at which something happens, often referred to as the "slope" of the function at a particular point.

Each term in a Taylor series depends on the derivatives of the function. Calculating these derivatives is often the first step in forming a Taylor polynomial. For our function \( f(x) = \frac{1}{1+x} \), here are the calculated derivatives:

• The first derivative \(f'(x) = -(1+x)^{-2}\) tells us about the linear change.
  
• The second derivative \(f''(x) = 2(1+x)^{-3}\) provides information about the rate of change of the slope.
  
• Third and higher derivatives like \(f'''(x) = -6(1+x)^{-4}\) and \(f^{(4)}(x) = 24(1+x)^{-5}\) delve into more complex changes.
  

Each derivative plays a role in forming terms of the Taylor series approximation, contributing to the polynomial's shape and accuracy.

polynomial approximation

Polynomial approximation is the essence of using Taylor polynomials. The idea is to simplify complex functions by approximating them with polynomials. These polynomials are easier to work with, analyze, and compute, making them a staple in calculus and applied mathematics.

In our example, the Taylor polynomial of degree 4 is an approximation of the function \(f(x)=\frac{1}{1+x}\). We derived it to be:

• \( P_4(x) = 1 - x + x^2 - x^3 + x^4 \)
  

This polynomial helps us understand the behavior of \(f(x)\) near \(x=0\) without dealing with the complexities of division by an expression. As the polynomial degree increases, its approximation of the function's behavior becomes more accurate.

The value in this approach is observed across many applications, from solving equations that can't be tackled directly to modeling scenarios in physics and engineering where exact values are either unnecessary or impractical to obtain.

function analysis

Analyzing a function involves understanding its behavior and properties across various regions. Taylor polynomials are a key part of this analysis, providing useful local approximations of functions.

In function analysis, we often need to consider how a function behaves as it approaches certain values—commonly zeros or infinity—or its behavior over specific intervals. By using the Taylor polynomial:

• We can predict the function’s behavior near a point of interest, such as checking if it has maximum or minimum values.
  
• They also aid in examining the function's continuity and smoothness; polynomials are continuously differentiable, providing smooth approximations.
  

Evaluating a function, like \( f(x) = \frac{1}{1+x} \), using its Taylor polynomial helps to render a clearer picture of its slope and curvature near \(x=0\). It’s these insights that are invaluable in both understanding theoretical concepts and applying mathematical techniques to real-world problems.