Problem 18 - Calculus with Concepts in Calculus

Question

Find all critical numbers of the given function. $$ f(x)=\ln \left(2 x+e^{-x}\right) $$

Step-by-Step Solution

Answer

The critical number is \( x = -\ln(2) \).

Step 1: Find the derivative

To find the critical numbers, we first need to find the derivative of the function. The given function is \[f(x)=\ln(2x + e^{-x}).\] Using the chain rule, the derivative is \[f'(x)= \frac{1}{2x + e^{-x}} \cdot (2 - e^{-x}).\] This simplifies to \[f'(x)= \frac{2 - e^{-x}}{2x + e^{-x}}.\]

Step 2: Set the derivative equal to zero

To find critical numbers, set the derivative equal to zero and solve for \(x\). \[ \frac{2 - e^{-x}}{2x + e^{-x}} = 0\]This implies that \[ 2 - e^{-x} = 0,\] which simplifies to \[ e^{-x} = 2.\]

Step 3: Solve for x

To solve \( e^{-x} = 2 \), take the natural logarithm of both sides: \[-x = \ln(2).\]This gives: \[ x = -\ln(2).\]

Step 4: Identify the critical number

The value \( x = -\ln(2) \) is a critical number for the function \( f(x) \), as it makes the derivative equal to zero.

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus that represent how a function changes as its input changes. Think of it as the slope of a function at a particular point. When we talk about the derivative of a function at a given point, we essentially measure how sensitive the function's value is to changes in its input around that point.

Some key points to keep in mind about derivatives include:

• The derivative of a function, noted as \( f'(x) \), gives us the rate at which \( f(x) \) changes with respect to x.
• The process of finding the derivative is known as differentiation.
• In practical terms, the derivative can tell us about the function's increasing or decreasing behavior and can help identify maximum and minimum points.

For the function \( f(x)=\ln(2x + e^{-x}) \), we used the derivative to find critical numbers. Critical numbers occur where the derivative is zero or undefined, indicating potential maxima, minima, or inflection points. Differentiation helps us to study the behavior and characteristics of functions more deeply.

Applying the Chain Rule

The chain rule is a vital tool in calculus used to differentiate composite functions—that is, functions made up by combining two or more functions. The simple idea behind the chain rule is that if a function can be expressed as a composition of other functions, its derivative can be found using the derivatives of those component functions.

In mathematical terms, if a function \( y = g(f(x)) \), where both \( f(x) \) and \( g(x) \) are differentiable, then the derivative \( y' \) can be given by \( g'(f(x)) \cdot f'(x) \). The rule allows us to handle otherwise complex derivatives efficiently.

• We apply the chain rule whenever a function is inside another function.
• The chain rule requires us to multiply the derivative of the outside function by the derivative of the inside function.

In our example, the function \( f(x)=\ln(2x+e^{-x}) \) uses the chain rule. Differentiating \( \ln(u) \), which is the outer function, along with differentiating \( 2x + e^{-x} \), the inner function, allows us to correctly find the overall derivative.

Natural Logarithm Insights

The natural logarithm, represented as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \) is approximately 2.71828. It's called "natural" because it appears in many natural growth processes and in solutions to problems involving exponential growth.

The natural logarithm has unique properties that simplify the calculation and manipulation:

• The natural logarithm of a product is the sum of the logarithms: \( \ln(ab) = \ln(a) + \ln(b) \).
• The natural logarithm of a quotient is the difference of the logarithms: \( \ln(a/b) = \ln(a) - \ln(b) \).
• The natural logarithm of a power is the exponent times the logarithm: \( \ln(a^b) = b\ln(a) \).

For the function \( f(x)=\ln(2x + e^{-x}) \), understanding these properties helps simplify the process of differentiation, particularly when applying the chain rule. The function \( 2x + e^{-x} \) gets plugged into the natural logarithm, creating a composite function that requires careful consideration when finding its derivative. Mastering the natural logarithm is crucial to dealing with exponential functions and their growth patterns effectively.