Problem 12 - Calculus for Biology and Medicine

Question

Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function 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 value is approximately 0.911, very close to the actual function value 0.90909 at \( x=0.1 \).

Step 1: Understanding the Function

The given function is \( f(x) = \frac{1}{1+x} \). This function can be expanded into a Taylor Series around \( x=0 \) which takes the form \( f(x) = \sum_{k=0}^{fty} (-1)^k x^k \).

Step 2: Write the Taylor Series Expansion

For the function \( f(x) = \frac{1}{1+x} \), the Taylor series expansion about \( x=0 \) is \( 1 - x + x^2 - x^3 + \dots \).

Step 3: Compute the Taylor Polynomial of Degree 3

The Taylor polynomial of degree 3 for \( f(x) = \frac{1}{1+x} \) is found by truncating the series after the \( x^3 \) term: \( P_3(x) = 1 - x + x^2 - x^3 \).

Step 4: Evaluate the Taylor Polynomial at \( x=0.1 \)

Substitute \( x=0.1 \) into the polynomial: \( P_3(0.1) = 1 - 0.1 + 0.1^2 - 0.1^3 = 1 - 0.1 + 0.01 - 0.001 = 0.911 \).

Step 5: Compute the Actual Function Value at \( x=0.1 \)

Calculate \( f(0.1) = \frac{1}{1+0.1} = \frac{1}{1.1} \approx 0.90909 \).

Step 6: Compare Values

The Taylor polynomial value \( P_3(0.1) \approx 0.911 \) and the actual function value \( f(0.1) \approx 0.90909 \) are very close. The polynomial gives a good approximation of the actual function.

Key Concepts

Taylor Series

A Taylor Series is a powerful mathematical tool that allows us to represent complex functions as infinite sums of simpler polynomial terms. This series is named after the mathematician Brook Taylor. It provides a way to approximate functions using derivatives evaluated at a single point, usually around 0, which is known as a Maclaurin series if expanded about zero.

• The general form of a Taylor Series for a function \( f(x) \) about the point \( x = a \) is given by:

\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots\]

• Each term involves higher-order derivatives of \( f \) evaluated at \( a \).
• By accounting for more terms in this series, the approximation becomes more accurate.

Understanding Taylor Series simplifies the process of evaluating complex expressions and helps solve calculus-based problems more efficiently.

Approximation

Approximation is a crucial concept in mathematical computations and analysis. It involves finding a close but not exact value of a function using simpler forms like polynomials. The degree of accuracy in an approximation depends on the number of terms used in a series.

• In the context of the Taylor Series, the degree of the polynomial that terminates the series defines its approximation level.
• For example, in our exercise, a Taylor polynomial of degree 3 is used to approximate \( f(x) = \frac{1}{1+x} \).
• The polynomial provides a result of \( P_3(0.1) \approx 0.911 \), which is notably close to the actual function value \( f(0.1) \approx 0.90909 \).

Through well-chosen approximations, complex real-world problems can be simplified, making them easier to solve with a margin of error within acceptable limits.

Calculus

Calculus is a fundamental branch of mathematics that studies change and motion, focusing on derivatives, integrals, and limits. It provides tools for analyzing and solving problems in science and engineering. Taylor Series and approximations are directly related to concepts from calculus.

• Derivatives, a core part of calculus, are used to create Taylor polynomials, which approximate functions.
• Calculus helps in understanding how small changes in inputs affect outputs, which is essential for accuracy when building approximations.
• The concept of limits within calculus also plays a significant role in convergence of Taylor Series, ensuring that as more terms are added, the polynomial approaches the function more closely.

With calculus as a toolbox, capturing a function's essence with its Taylor Series expansion becomes a strategic way to tackle both theoretical and practical problems.