Problem 12 - Calculus for Biology and Medicine

Question

In Problems 11-16, compute the Taylor polynomial of degree \(n\) about \(a=0\) for the indicated functions and compare the value of the functions at the indicated point with the value of the corresponding Taylor polynomial. $$ f(x)=\frac{1}{1-x}, n=3, x=0.1 $$

Step-by-Step Solution

Answer

The Taylor Polynomial is 1.111 and the function value is approximately 1.1111 at \( x = 0.1 \); they are close.

Step 1: Understanding the Function and Taylor Series

The function given is \( f(x) = \frac{1}{1-x} \). The Taylor series for \( f(x) \) at \( a = 0 \) is obtained by considering its geometric series form. The Taylor series is \( f(x) = 1 + x + x^2 + x^3 + \ldots \).

Step 2: Extracting the Taylor Polynomial of Degree 3

The third-degree Taylor polynomial, \( P_3(x) \), will include terms up to \( x^3 \). For \( f(x) = \frac{1}{1-x} \), this gives \( P_3(x) = 1 + x + x^2 + x^3 \).

Step 3: Evaluating the Taylor Polynomial at x = 0.1

To find \( P_3(0.1) \), plug \( x = 0.1 \) into the polynomial: \( P_3(0.1) = 1 + 0.1 + 0.1^2 + 0.1^3 \). This equals \( 1 + 0.1 + 0.01 + 0.001 = 1.111 \).

Step 4: Evaluating the Function at x = 0.1

Find \( f(0.1) \) by substituting \( x = 0.1 \) into the original function: \( f(0.1) = \frac{1}{1-0.1} = \frac{1}{0.9} \approx 1.1111 \).

Step 5: Comparing the Function and Polynomial Values

The value of the Taylor polynomial \( P_3(x) \) at \( x = 0.1 \) is 1.111. The value of the function \( f(x) \) at the same point is approximately 1.1111. Both values are quite close, showing that the polynomial is a good approximation at this point.

Key Concepts

Taylor Polynomial

A Taylor Polynomial is a type of polynomial used to approximate a function near a specific point, known as the center point. This polynomial is generated from the Taylor series, which is an infinite series expansion of the function. In the case of the original exercise, the function is approximated by its Taylor series centered at point 0, and we use a polynomial of degree 3.

• This involves truncating the series after a limited number of terms to form the polynomial.
• The degree of the polynomial (in this case, 3) indicates the highest power of x included in the polynomial.

Using Taylor Polynomials, we can approximate functions to a desired accuracy while only considering a finite number of terms, which is highly practical when calculating by hand or using limited computational resources.

Geometric Series

The Geometric Series is fundamental to understanding how the Taylor Polynomial for our function is formed. A geometric series has the form:\[ a + ar + ar^2 + ar^3 + \ldots \] where 'a' is the first term, and 'r' is the common ratio.

• In this exercise, the Taylor series for the function \( f(x) = \frac{1}{1-x} \) can be expressed as a geometric series.
• Here, the first term is 1, and the common ratio is \( x \).

The geometric nature of this series allows us to easily derive the polynomial by summing a limited number of terms, making use of its converging property.

Function Approximation

Function Approximation is an essential concept in calculus that involves finding a simpler expression that approximates a more complex function. This is often done using Taylor Polynomials. The step-by-step solution demonstrates how closely a third-degree Taylor Polynomial can approximate the function \( f(x) = \frac{1}{1-x} \) around \( x = 0.1 \).

• By comparing the Taylor polynomial \( P_3(0.1) = 1.111 \) and the actual value \( f(0.1) \approx 1.1111 \), we see the approximation is very close.
• Such approximation techniques are useful in various applications, including numerical analysis, computer science, and engineering.

Hence, we can achieve high accuracy with a relatively low-degree polynomial for smooth and continuous functions when evaluated close to the center point.

Degree of Polynomial

The degree of a polynomial tells us the highest power of x that appears in the polynomial. It's a primary factor in determining the accuracy and complexity of a Taylor Polynomial.
For the Taylor polynomial of degree 3 in our series, this means:

• We include terms from degree 0 up to degree 3, i.e., 1, \( x \), \( x^2 \), and \( x^3 \).
• Each higher degree polynomial will incorporate additional terms, improving the function approximation when more precision is desired.

While higher degree polynomials generally result in better approximations, they are also computationally more taxing, demanding careful balance between simplicity and accuracy.