Problem 54 - Calculus and its applications

Question

Given that the derivative of \(f(x)=a^{x}\) is \(f^{\prime}(x)=a^{x}(\ln a),\) in Section 3.1 we showed that \(f^{\prime}(x)=a^{x} \cdot \lim _{h \rightarrow 0} \frac{a^{h}-1}{h}\). Thus, we can define \(\ln a=\lim _{h \rightarrow 0} \frac{a^{h}-1}{h}\). Use this definition to find each limit. $$ \lim _{h \rightarrow 0} \frac{3^{h}-1}{h} $$

Step-by-Step Solution

Answer

\( \ln 3 \approx 1.0986 \).

Step 1: Understand the Problem

We are tasked with finding the limit \( \lim_{h \to 0} \frac{3^{h} - 1}{h} \). This limit helps determine the natural logarithm of 3 as it is analogous to the definition: \( \ln a = \lim _{h \to 0} \frac{a^{h}-1}{h} \). Thus, we need to compute this limit to find \( \ln 3 \).

Step 2: Apply the Known Definition

Using the known formula for the derivative and the definition of natural logarithm, we can directly apply that to our specific function where \( a = 3 \). According to the definition given, \( \ln 3 = \lim _{h \to 0} \frac{3^{h}-1}{h} \).

Step 3: Evaluate the Limit

For practical calculation, the limit \( \lim _{h \to 0} \frac{3^{h}-1}{h} \) can be approached numerically or conceptually. As this matches the definition of \( \ln 3 \), computing this directly will give us \( \ln 3 \).

Step 4: Substitute Known Values

We recognize that by definition, this limit equals \( \ln 3 \). The value of \( \ln 3 \) is a known constant approximately equal to 1.0986.

Key Concepts

Natural Logarithm

The concept of the natural logarithm is essential in calculus, especially when dealing with exponential functions. The natural logarithm, denoted as \(\ln\), is the inverse operation of the exponential function with base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. When we apply the natural logarithm to an exponential, it essentially "undoes" the exponential.

• For any positive number \(a\), \(\ln a\) is the power to which \(e\) must be raised to result in \(a\).
• This makes it particularly useful in solving equations involving exponentials, as it helps isolate the variable.
• In the context of calculus, the natural logarithm comes in handy while dealing with the derivatives of exponential functions.

Understanding \(\ln a\) through its limit definition, as defined in this exercise, roots it deeply in the foundations of calculus. It is always crucial when evaluating the behavior of functions at very small scales, such as near zero, which is often tackled using limits.

Limit Definition

The limit definition is a fundamental piece in understanding calculus and how it operates, especially in defining derivatives and integrals. In simplest terms, a limit helps us find out what value a function tends to as the input approaches some number.

• The expression \(\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\) is used to define derivatives, as it tells us how the function \(f(x)\) changes as we make a very small change \(h\) in \(x\).
• Applying the limit definition to exponential functions allows us to derive their behavior and properties, such as the natural logarithm in our exercise.

In the exercise, we used the expression \(\lim_{h \to 0} \frac{3^h - 1}{h}\) to determine \(\ln 3\). By seeing how the fraction \(\frac{3^h - 1}{h}\) behaves as \(h\) becomes very small, we derive \(\ln 3\). Thus, limits help us translate a function's local behavior into precise numeric values, as demonstrated by finding \(\ln 3\).

Calculus Concepts

Calculus concepts provide a framework to analyze changes and motion, unlocking a deeper understanding of the world around us. Two of the main pillars of calculus are differentiation and integration. In differentiation, we measure how a function changes, while integration deals with accumulating quantities.

• The derivative of an exponential function, for example, is expressed using the base of the exponential times the natural logarithm of the base, which offers insights into the rate of change.
• Exponential functions are prevalent in growth and decay models, making their derivatives crucial in modeling real-life scenarios such as population growth, radioactive decay, and financial calculations.

Mastering calculus concepts like limits, derivatives, and integrals allows for solving complex problems. Through exercises like finding \(\ln 3\), involving derivatives and limits, we see calculus principles applied to compute precise outcomes, enhancing our analytical tools for both theoretical and practical applications.