Problem 16 - 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. $$ f(x)=\ln (1+x), n=3, x=0.1 $$

Step-by-Step Solution

Answer

The third degree Taylor polynomial is \( P_3(x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \), and \( P_3(0.1) \approx 0.0950333 \) closely approximates \( \ln(1.1) \approx 0.0953 \).

Step 1: Understand the Function and Taylor Series

We need to find the third degree Taylor polynomial for the function \( f(x) = \ln (1+x) \) about \( a = 0 \). The formula for the Taylor series expansion is \( P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k \), where \( f^{(k)}(a) \) is the \( k \)-th derivative of the function evaluated at \( a \).

Step 2: Calculate Derivatives

Calculate the first few derivatives of \( f(x) = \ln(1+x) \):- \( f(x) = \ln(1+x) \)- \( f'(x) = \frac{1}{1+x} \)- \( f''(x) = -\frac{1}{(1+x)^2} \)- \( f'''(x) = \frac{2}{(1+x)^3} \).Evaluate these at \( a = 0 \).

Step 3: Evaluate Derivatives at \( a = 0 \)

Evaluate the derivatives from Step 2 at \( x = 0 \):- \( f(0) = \ln(1+0) = 0 \)- \( f'(0) = \frac{1}{1+0} = 1 \)- \( f''(0) = -\frac{1}{(1+0)^2} = -1 \)- \( f'''(0) = \frac{2}{(1+0)^3} = 2 \).

Step 4: Formulate the Taylor Polynomial

Use the derivatives to form the Taylor polynomial of degree 3:\[ P_3(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 \]Substitute the values:\[ P_3(x) = 0 + 1 \cdot x - \frac{1}{2}x^2 + \frac{2}{6}x^3 = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \].

Step 5: Compute \( f(0.1) \) and \( P_3(0.1) \)

Calculate the actual function value and polynomial value at \( x = 0.1 \):- \( f(0.1) = \ln(1+0.1) = \ln(1.1) \approx 0.0953 \)- \( P_3(0.1) = 0.1 - \frac{1}{2}(0.1)^2 + \frac{1}{3}(0.1)^3 = 0.1 - 0.005 + 0.0000333 \approx 0.0950333 \).

Step 6: Compare Values

Compare the computed values:- \( f(0.1) \approx 0.0953 \)- \( P_3(0.1) \approx 0.0950333 \).The Taylor polynomial \( P_3(0.1) \) is close to the actual value \( f(0.1) \), demonstrating that the polynomial provides a good approximation for small \( x \).

Key Concepts

Taylor series

Taylor series is a fascinating topic in mathematics that helps us express functions as infinite sums of terms calculated from the values of a function’s derivatives at a single point. This single point, usually denoted as \(a\), is around which the series is expanded. The beauty of the Taylor series lies in its ability to approximate complex functions using relatively simple polynomial equations.

The general formula for the Taylor series of a function \( f(x) \) about a point \( a \) is given by:

• \[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k \]

Here, \( f^{(k)}(a) \) is the \( k \)-th derivative of \( f(x) \) evaluated at \( a \), and \( k! \) is the factorial of \( k \). Each term in the series adds another layer of precision to the approximation, with higher-degree terms becoming significant as \( x \) moves farther from \( a \). This infinite series of terms makes Taylor series applicable in numerous fields, from physics to finance.

Understanding Taylor series can significantly facilitate working with calculus, especially when it comes to approximating functions and solving differential equations.

calculus

Calculus, known for its two fundamental branches - differential calculus and integral calculus, forms the cornerstone of modern mathematics. These two areas focus on the concepts of change and accumulation, forming the backbone of many scientific advancements.

In relation to Taylor series, differential calculus plays a crucial role. It involves the study of derivatives, which are essentially measures of how a function changes as its input changes. Understanding derivatives is essential for constructing Taylor series because the Taylor polynomial's terms incorporate various derivatives of the function.

For example, in the function \( f(x) = \ln(1+x) \), derivatives such as \( f'(x) = \frac{1}{1+x} \) and \( f''(x) = -\frac{1}{(1+x)^2} \) are calculated, and these derivatives are then evaluated at a specific point \( a \), which in this case is 0. Utilizing calculus, we can derive a polynomial that closely approximates the behavior of a more complicated function.

approximation

Approximation is the art of finding a simple, close, yet not exact value or representation for something complex. In mathematics, approximation is critical when the exact values are challenging to compute or unnecessary for practical purposes.

Taylor polynomials are an important tool for approximation. They provide polynomial expressions that approximate more complex functions within a certain range of values. For instance, instead of calculating \( \ln(1.1) \) directly, which may not always be straightforward, you can use the third-degree Taylor polynomial to approximate it efficiently.

In the context of our example:

• The third-degree Taylor polynomial is \( P_3(x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \).
• When \( x = 0.1 \), computing \( P_3(0.1) \approx 0.0950333 \) closely estimates \( \ln(1.1) \approx 0.0953 \).

In cases of small \( x \), where the actual function value and polynomial estimation are very similar, Taylor series offer a convenient method for fast approximations.

degree of polynomial

The degree of a polynomial is the highest power of the variable in its expression. In a Taylor series, the degree of the polynomial controls the number of terms used in the approximation.

A Taylor polynomial of degree \( n \) includes terms from 0 up to \( n \). This means that the more terms (higher degree), the more accurate the approximation becomes. For example, a third-degree polynomial uses up to the cubic term, providing a balance between simplicity and accuracy for small values of \( x \).

In our exercise with \( f(x) = \ln(1+x) \), the third-degree Taylor polynomial is expressed as:

• \[ P_3(x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \]

Including up to the third power of \( x \), correctly captures the function's behavior near \( a = 0 \). Hence, understanding the degree of polynomial helps realize how precisely we can approximate a function using its Taylor polynomial, making calculus more versatile and applicable to a broader range of problems.