Problem 55 - 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. $$ \lim _{h \rightarrow 0} \frac{5^{h}-1}{h} $$

Step-by-Step Solution

Answer

The limit is \(\ln 5\).

Step 1: Identify the Function and Its Derivative

We are given the function \(f(x) = 5^x\). According to the problem statement and the provided derivative formula, the derivative of this function is given as \(f'(x) = 5^x \ln 5\).

Step 2: Recall the Limit Definition of the Natural Logarithm

We know that \(\ln a = \lim_{h \rightarrow 0} \frac{a^h - 1}{h}\). The problem is asking us to find \(\lim_{h \rightarrow 0} \frac{5^h - 1}{h}\). Comparing this with the definition, we see that \(a = 5\).

Step 3: Apply the Definition to Find the Limit

Using the known relationship, we set \(a = 5\) and directly apply the definition: \[\lim_{h \rightarrow 0} \frac{5^h - 1}{h} = \ln 5.\] This shows the result of this limit as \(\ln 5\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. A classic form of an exponential function is \( a^x \), with \( a \) being the base and \( x \) denoting the exponent which varies. Exponential functions are distinct from polynomial functions by this constant base aspect. Moreover, they grow at a much faster rate when their exponent is positive.

• Growth and Decay: When \( a > 1 \), the function experiences exponential growth as \( x \) increases. In contrast, when \( 0 < a < 1 \), the function undergoes exponential decay.
• Application Examples: These functions model scenarios where growth or decay rates depend on the current value, like in populations or radioactive decay.

The derivative of an exponential function, as given in the exercise, highlights how these functions are differentiated. For \( f(x) = a^x \), the derivative is \( f'(x) = a^x \ln a \), which depends on the function itself and the natural logarithm of the base.

Limits

Limits play a crucial role in calculus, serving as the foundation for defining derivatives and integrals. A limit essentially describes the behavior of a function as its input approaches a particular value but doesn’t necessarily reach it. Understanding limits allows us to handle values that tend toward infinity or indeterminate forms.

• Basic Idea: The limit of a function \( f(x) \) as \( x \) approaches some value \( c \) is the value that \( f(x) \) gets closer to, whether right from its left or from its right.
• Notation: The limit is denoted by \( \lim_{x \to c} f(x) \).
• Practical Use: Limits help in defining the notion of derivatives; for example, the derivative at a point often involves limits as in the formula: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

In the provided exercise, limits are used to define natural logarithms, showing how limits and logarithmic functions are intertwined.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a significant mathematical function. It is the logarithm to the base \( e \), where \( e \approx 2.71828 \), known as Euler's number. The natural logarithm has several important properties and plays a key role in calculus and exponential functions.

• Definition: The natural logarithm of a number is the exponent to which \( e \) must be raised to yield that number. For instance, \( \ln e = 1 \), as \( e^1 = e \).
• Derivative: The derivative of \( \ln x \) is \( \frac{1}{x} \), showcasing one reason why natural logs are useful in calculus.
• Logarithm Properties: These include that \( \ln(ab) = \ln a + \ln b \) and \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).

In the exercise, the concept of natural logarithms is employed to find specific limits related to exponential functions, solidifying their connection in calculus.