Problem 18 - Calculus for Biology and Medicine

Question

Approximate \(f(x)\) at a by the linear approximation $$L(x)=f(a)+f^{\prime}(a)(x-a)$$ $$ f(x)=\ln (1+2 x) \text { at } a=0 $$

Step-by-Step Solution

Answer

The linear approximation of \( f(x) = \ln(1+2x) \) at \( a=0 \) is \( L(x) = 2x \).

Step 1: Identify Components of the Linear Approximation Formula

The formula for linear approximation is given by \( L(x) = f(a) + f'(a)(x-a) \). For this problem, we are to approximate \( f(x) = \ln(1+2x) \) at \( a=0 \). This means we need to calculate \( f(a) \) and \( f'(a) \) at \( a=0 \).

Step 2: Calculate \( f(a) \)

Substitute \( a = 0 \) into the function \( f(x) = \ln(1+2x) \). Hence, \( f(0) = \ln(1+2 \cdot 0) = \ln(1) = 0 \).

Step 3: Differentiate the Function

To find \( f'(x) \), differentiate \( f(x) = \ln(1+2x) \) with respect to \( x \). Using the chain rule, we get \( f'(x) = \frac{1}{1+2x} \cdot 2 = \frac{2}{1+2x} \).

Step 4: Evaluate \( f'(a) \) at \( a=0 \)

Now, substitute \( a = 0 \) into the derivative \( f'(x) = \frac{2}{1+2x} \). Thus, \( f'(0) = \frac{2}{1+2 \cdot 0} = 2 \).

Step 5: Formulate the Linear Approximation

Now that we have \( f(0) = 0 \) and \( f'(0) = 2 \), substitute these into the linear approximation formula: \( L(x) = f(0) + f'(0)(x-0) \). Therefore, \( L(x) = 0 + 2x \), simplifying to \( L(x) = 2x \).

Key Concepts

Understanding Derivatives

At the heart of linear approximation lies the concept of derivatives. A derivative represents the rate at which a function changes at a specific point. It's akin to finding the slope of a tangent line to a curve at a given point. In mathematical terms, the derivative of a function, often expressed as \( f'(x) \), gives insight into the function's behavior and change rate at any point \( x \).

When approximating a function linearly, we rely on its derivative because it reflects how the function increases or decreases. For instance, in our exercise, differentiating \( f(x) = \ln(1+2x) \) leads us to the derivative \( f'(x) = \frac{2}{1+2x} \). This derivative helps us estimate the function's behavior near \( x = 0 \).

The Process of Differentiation

Differentiation is a key technique in calculus used to find a derivative. It involves applying specific rules to a function to determine its derivative. For complex functions, rules like the chain rule become essential.

In the given exercise, we differentiate \( f(x) = \ln(1+2x) \) using the chain rule. The chain rule helps us handle composite functions, where one function is nested inside another. If we see \( u = 1+2x \), then \( f(x) = \ln(u) \) and differentiating gives \( f'(x) = \frac{1}{u} \cdot \frac{du}{dx} = \frac{2}{1+2x} \). This differentiation step provides the necessary slope for our linear approximation close to \( a=0 \).

Exploring the Natural Logarithm

The natural logarithm, symbolized as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \approx 2.718 \). It is a fundamental analytical function used to model growth or decay processes. Understanding its properties aids in various calculus operations, such as differentiation.

In our exercise, the function \( f(x) = \ln(1+2x) \) focuses on this natural logarithm property. Evaluating the natural logarithm at specific points gives us key values for linear approximations. Here, \( \ln(1) = 0 \), simplifying our calculations at \( a=0 \). Being familiar with these logarithmic identities and their behavior assists in making accurate linear approximations of functions.