Problem 20 - Calculus for Biology and Medicine

Question

Compute the Taylor polynomial of degree \(n\) about a and compare the value of the approximation with the value of the function at the given point \(x\). $$ f(x)=\ln x, a=1, n=3 ; x=2 $$

Step-by-Step Solution

Answer

The approximation \(P_3(2) = \frac{5}{6}\), which is approximately 0.833, overestimates the actual value of \(\ln 2\approx 0.693\).

Step 1: Identify Taylor Polynomial Formula

The Taylor polynomial of degree \(n\) for a function \(f(x)\) about \(a\) is given by: \[P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \ldots\] For \(n=3\), we will use the terms up to \(f'''(a)\).

Step 2: Compute Derivatives

Compute the derivatives of \(f(x) = \ln x\):- \(f(x) = \ln x\)- \(f'(x) = \frac{1}{x}\)- \(f''(x) = -\frac{1}{x^2}\)- \(f'''(x) = \frac{2}{x^3}\)

Step 3: Evaluate Derivatives at \(a = 1\)

Substitute \(a = 1\) into the derivatives:- \(f(1) = \ln 1 = 0\)- \(f'(1) = \frac{1}{1} = 1\)- \(f''(1) = -\frac{1}{1^2} = -1\)- \(f'''(1) = \frac{2}{1^3} = 2\)

Step 4: Construct Taylor Polynomial

Substitute the values into the Taylor polynomial formula:\[P_3(x) = 0 + 1(x - 1) + \frac{-1}{2}(x - 1)^2 + \frac{2}{6}(x - 1)^3\]Simplify:\[P_3(x) = (x - 1) - \frac{1}{2}(x - 1)^2 + \frac{1}{3}(x - 1)^3\]

Step 5: Compute \(P_3(2)\)

Substitute \(x = 2\) into the polynomial:\[P_3(2) = (2 - 1) - \frac{1}{2}(2 - 1)^2 + \frac{1}{3}(2 - 1)^3\]Evaluate:\[P_3(2) = 1 - \frac{1}{2}(1) + \frac{1}{3}(1)\]Simplify:\[P_3(2) = 1 - \frac{1}{2} + \frac{1}{3} = \frac{6}{6} - \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\]

Step 6: Compare with \(f(2)\)

Compute the actual value of \(f(2)\):\[f(2) = \ln 2\approx 0.693\]The approximation \(P_3(2) = \frac{5}{6} \approx 0.833\) is greater than actual \(f(2)\approx 0.693\).

Key Concepts

Derivatives

Derivatives are fundamental in the study of calculus, and they have a pivotal role in the construction of Taylor polynomials. A derivative represents the rate at which a function is changing at any given point. It provides us with insights into the behavior of the function, such as identifying minima, maxima, and points of inflection.
For a function like the natural logarithm, denoted as \( f(x) = \ln x \), the derivatives help us understand its curve dynamics. Here's a quick overview of the first few derivatives:

• First derivative: \( f'(x) = \frac{1}{x} \) – This tells us how steep the curve is at any point \(x\).
• Second derivative: \( f''(x) = -\frac{1}{x^2} \) – A negative value indicates concavity or the 'bending' direction of the curve.
• Third derivative: \( f'''(x) = \frac{2}{x^3} \) – This can be used to refine our approximation further.

When we calculate these derivatives at \( a = 1 \), they give us the increments used in our Taylor polynomial for \( \ln x \). These derivations are essential for estimating the curve at points close to \( x = 1 \).

Approximation

The concept of approximation lies at the heart of Taylor polynomials. In mathematics, especially calculus, it's often useful to approximate complex functions using polynomials because they are easier to work with. Taylor polynomials serve as powerful tools to represent functions as sums of derivatives evaluated at a specific point \( a \).
Here, a Taylor polynomial approximates a function like \( f(x) = \ln x \) around a chosen point, and we calculated up to the third degree. This involves summing up to the third term of the Taylor series equation:

• \( P_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 \)

For \( n = 3 \), our Taylor polynomial provides a degree of accuracy for estimating the function's value near \( a = 1 \). The approximation shows that as we move from \( x = 1 \) to \( x = 2 \), the Taylor polynomial captures the curve's trajectory better with increasing degrees.

Logarithmic Function

The natural logarithm function, denoted \( f(x) = \ln x \), is one of the most important logarithmic functions you'll encounter in mathematics. It has unique properties and essential applications across various fields such as engineering, physics, and finance.
In the context of Taylor polynomials, the function's nature necessitates approximation, especially since its exact behavior is difficult to describe in polynomial form. The function \( \ln x \) grows slower as \( x \) increases, and it passes through \( (1, 0) \) because \( \ln 1 = 0 \).
Logarithmic functions are beneficial for modeling growth processes or decay, where rates of change evolve over time. Through the step-by-step derivation using Taylor's approximation, we achieved an expression for \( \ln x \) near \( a = 1 \) and used it to estimate at \( x = 2 \). This demonstrates how Taylor polynomials simplify analyzing otherwise complex functions, yet they remain as approximations and not exact representations.