Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

a. Compute the fourth degree polynomial, $$ p(x)=\pi_{0}+\pi_{1}(x-1)^{2}+\pi_{2}(x-1)^{2}+\pi_{3}(x-1)^{3}+\pi_{4}(x-1)^{4} $$ that matches \(f(x)=\ln x\) at the anchor point \(a=1\). b. From the pattern of coefficients in \(p\), guess the fifth degree term of a fifth degree polynomial that matches \(f(x)=\ln x\) at the anchor point \(a=1\)

Step-by-Step Solution

Answer

Compute the coefficients by differentiating \( \ln x \) and using Taylor series expansion.

Step 1: Understand the polynomial structure

The given polynomial is of fourth degree, expressed with coefficients \( \pi_0, \pi_1, \pi_2, \pi_3, \text{ and } \pi_4 \). It uses the form \( p(x) = \pi_0 + \pi_1(x-1)^2 + \pi_2(x-1)^2 + \pi_3(x-1)^3 + \pi_4(x-1)^4 \). This expression needs to represent the function \( f(x) = \ln x \) at the anchor point \( a = 1 \).

Key Concepts

Logarithmic Function

A logarithmic function is a mathematical operation that is the inverse of the exponential function. It is defined as the power to which a base, often 10 or the natural base e, must be raised to produce a certain number. In this case, the function we are focusing on is \(f(x) = \ln x\).\\This particular logarithmic function is known as the natural logarithm, where the base is the mathematical constant \(e \), approximately equal to 2.718. The natural logarithm of a number provides insights into growth and decay processes, as it frequently appears in problems related to population growth, radioactive decay, and finance.\\When working with logarithms in calculus, the derivatives become key players—for instance, the derivative of \(\ln x\) is \(\frac{1}{x}\). This fact is critical when creating polynomial approximations, as it helps determine how the polynomial should change to best mimic the logarithmic function.

Polynomial Approximation

Polynomial approximation is a technique in calculus where we estimate more complex functions using polynomials, which are expressions consisting of variables and coefficients. These approximations help simplify computations while maintaining a reasonable accuracy, especially around the point of interest.\\In the case of Taylor Polynomial, it breaks down a function into an infinite series of terms. Each term consists of derivatives of the function evaluated at a specific point and then multiplied by a certain power of \(x - a\). This is helpful because polynomials are easier to work with compared to functions like \(\ln x\).\\For the exercise, we use polynomial approximation to create a polynomial, \(p(x)\), that approximates \(f(x)=\ln x\) near \(a = 1\). The polynomial is constructed by finding coefficients \((\pi_0, \pi_1, \ldots)\) such that the derivatives of \(\ln x\) and \(p(x)\) coincide at the anchor point. This ensures \(p(x)\) behaves very similarly to \(\ln x\) close to that chosen point.

Degree of Polynomial

The degree of a polynomial refers to the highest power of the variable \(x\) that appears with a non-zero coefficient. It is an essential concept in understanding the polynomial's behavior and complexity.\\In our problem, we deal with a fourth-degree polynomial, \(p(x)\). This means that the highest power of \(x\) is four, which defines the overall shape and the nature of \(p(x)\). Increasing the degree of the polynomial generally improves the approximation to \(f(x)=\ln x\), especially in a wider range around the anchor point \(a=1\).\\While working with Taylor polynomials, the degree corresponds to the number of terms used from the Taylor series. In part b of the exercise, you're asked to extend to the fifth-degree term, implying an additional precision step by predicting what the polynomial would look like if we added another term. This involves identifying patterns in previously determined coefficients and using insights from calculus to guess the next term. This allows us to better tailor the polynomial to the particularities of \(\ln x\).