Problem 48 - Calculus Single Variable

Question

Specify a function \(f\) and a value \(c\) for which the given limit equals \(f^{\prime}(c) .\) (You need not evaluate the limit.) $$ \lim _{h \rightarrow 0} \frac{10^{h}-1}{h} $$

Step-by-Step Solution

Answer

\( f(x) = 10^x \) at \( c = 0 \).

Step 1: Identify the limit form

The given limit is \( \lim_{h \to 0} \frac{10^h - 1}{h} \). This resembles the definition of the derivative of a function at a point \( c \), specifically \( f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \).

Step 2: Define function \( f \)

In order to match the form of \( f'(c) \), identify a function \( f(x) = 10^x \), because \( f(h) = 10^h \) appears in the numerator of the limit.

Step 3: Determine point \( c \)

To make the expression exactly match \( f'(c) \), consider \( f(c) = 10^c = 1 \). This corresponds to \( c = 0 \) because \( 10^0 = 1 \). Hence, we use \( c = 0 \).

Step 4: Confirm using the derivative definition

The limit \( \lim_{h \to 0} \frac{10^h - 1}{h} \) is in the form \( \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h} \) which confirms that it represents \( f'(0) \) for the function \( f(x) = 10^x \).

Key Concepts

Derivative Definition

Before diving into the specifics of the exercise, let's understand the basic definition of a derivative. A derivative, in simple terms, is a way to measure how a function changes as its input changes. It's the mathematical tool used to determine the slope of a curve at any given point.

The standard definition of the derivative of a function, denoted as \(f'(c)\), at a point \(c\) involves the concept of limits.

• The derivative \(f'(c)\) is calculated using:
  \[ f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \]
• This formula essentially captures the idea of finding the instantaneous rate of change, or the slope, at a specific point \(c\).
• The expression \(f(c+h) - f(c)\) represents the change in the function's output as the input changes from \(c\) to \(c+h\).

The challenge in the exercise asked us to identify a function and a specific value \(c\) such that the given limit expression fits this derivative formula. Understanding this definition is crucial for correctly interpreting problems involving derivatives.

Limit of a Function

The limit of a function is a foundational concept in calculus. It describes the value that a function approaches as the input approaches some value. In the context of derivatives, limits are essential for defining the derivative itself.

Here's a simple breakdown:

• A limit evaluates what happens as your input "gets close to" a particular value, rather than what happens when it actually equals that value.
• The expression \(\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}\) uses limits to analyze how the function \(f\) behaves as \(h\) approaches zero.
• This is because directly substituting \(h = 0\) in the expression \(\frac{f(c+h) - f(c)}{h}\) would result in a division by zero issue.

By understanding limits, students can comprehend how functions behave in an "almost" scenario, which is pivotal for grasping more advanced calculus topics like continuity and differentiability.

Exponential Functions

Exponential functions are an important class of functions in calculus. These functions, where the variable is an exponent, have distinct properties that make them extremely useful in real-world applications.

Understanding exponential functions involves knowing these core aspects:

• The general form of an exponential function is \(f(x) = a^x\), where \(a\) is a positive constant not equal to 1.
• For our problem, \(a\) is 10, so the function is \(f(x) = 10^x\).
• Such functions exhibit rapid growth or decay, depending on whether \(a\) is greater than or less than 1.

In the exercise, recognizing the function \(f(x) = 10^x\) in the limit was a key step toward understanding which function's derivative the limit represented.While exponential functions might seem daunting initially, they're foundational in modeling numerous natural phenomena, ranging from population growth to radioactive decay.