Problem 24 - Calculus for Biology and Medicine

Question

how that $$ T^{4} \approx T_{a}^{4}+4 T_{a}^{3}\left(T-T_{a}\right) $$ for \(T\) close to \(T_{a}\)

Step-by-Step Solution

Answer

Use Taylor series expansion to show that \( T^4 \approx T_a^4 + 4T_a^3(T - T_a) \) for \( T \) near \( T_a \).

Step 1: Understand the Problem

We need to show the approximation of the expression \( T^4 \) around \( T_a \) using a linear approximation method. The formula given is a version of the first-degree Taylor polynomial expansion around the point \( T_a \).

Step 2: Recall Taylor Series Expansion

The Taylor series expansion of a function \( f(T) \) around a point \( T_a \) is given by: \[ f(T) \approx f(T_a) + f'(T_a)(T - T_a) + \frac{f''(T_a)}{2!}(T - T_a)^2 + \ldots \]For a linear approximation, only the first two terms are needed.

Step 3: Apply the Taylor Expansion for \( T^4 \)

Consider the function \( f(T) = T^4 \).- First, calculate \( f(T_a) \):\[ f(T_a) = (T_a)^4 \]- Next, calculate the first derivative, \( f'(T) \):\[ f'(T) = 4T^3 \]- Evaluate this derivative at \( T = T_a \):\[ f'(T_a) = 4(T_a)^3 \]

Step 4: Substitute into the Taylor Expansion

Insert the calculated values into the first-degree Taylor expansion formula:\[ T^4 \approx (T_a)^4 + 4(T_a)^3 (T - T_a) \]This matches the formula given in the problem statement: \[ T^4 \approx T_a^4 + 4T_a^3(T - T_a) \]

Step 5: Complete the Analysis

The expression we derived using the Taylor expansion matches exactly with the given approximation. Thus, the approximation is correct for \( T \) close to \( T_a \).

Key Concepts

First-Degree Taylor Polynomial

To approximate a function near a specific point, we can use a Taylor polynomial. The first-degree Taylor polynomial is an elegant way to form a linear approximation of a function around a point. It considers only the immediate behavior of the function and its first derivative. The general form of a first-degree Taylor polynomial for a function \( f(T) \) about a point \( T_a \) is:

• \( f(T) \approx f(T_a) + f'(T_a)(T - T_a) \)

This captures the value of the function at \( T_a \) and its slope there, giving us a simple linear equation.This polynomial acts as a tangent line closely approximating \( f(T) \) when \( T \) is near \( T_a \). The closer \( T \) is to \( T_a \), the better the approximation will be. It can be quite useful for quick calculations in situations where a detailed function evaluation isn't necessary.

Derivative Calculation

Calculating derivatives is crucial for forming Taylor polynomials. The derivative gives us the slope of the function at any point. This is vital for approximations, as derivatives reflect how the function is changing.For the function \( f(T) = T^4 \), the derivative is obtained as follows:

• First Derivative: \( f'(T) = 4T^3 \)

This derivative tells us how \( T^4 \) changes as \( T \) changes.Evaluating this derivative at \( T = T_a \) provides the slope at this specific point:

• \( f'(T_a) = 4(T_a)^3 \)

This slope helps form the linear term in the first-degree Taylor polynomial.Understanding how to calculate and interpret derivatives is essential for building approximations and exploring function behavior.

Function Approximation

Function approximation is a powerful tool in mathematics that allows us to simplify complex functions into more manageable forms. Using Taylor polynomials, particularly first-degree, we linearize these functions for practical and efficient use.When approximating \( T^4 \) near \( T_a \), the first-degree Taylor polynomial gives:

• \( T^4 \approx T_a^4 + 4T_a^3(T - T_a) \)

Here, the approximation shows the function value and the immediate change around \( T_a \).This approach is particularly helpful when exact values are challenging to compute, or when performing calculations where precise values are not necessary. By focusing on local behavior, this approximation gives a snapshot of how the function behaves close to \( T_a \), making difficult problems more approachable.