Problem 34 - Brief Applied Calculus

Question

Find \(f^{\prime}(x)\) by using the definition of the derivative. [Hint: See Example 4.] $$ \underline{\phantom{xxx}} f(x)=\pi $$

Step-by-Step Solution

Answer

The derivative \( f'(x) \) is 0 because \( f(x) = \pi \) is a constant function.

Step 1: Recall the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x \) is given by the limit: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \] This formula finds the instantaneous rate of change of the function at any point \( x \).

Step 2: Apply the Function to the Formula

Given the function \( f(x) = \pi \), substitute into the derivative formula: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{\pi - \pi}{h} \]

Step 3: Simplify the Expression

The expression becomes:\[ f'(x) = \lim_{h \to 0} \frac{0}{h} = \lim_{h \to 0} 0 \]Since the numerator is 0, the entire expression is 0 regardless of \( h \).

Step 4: Evaluate the Limit

When \( h \to 0 \), the expression remains as 0. Thus, the derivative is:\[ f'(x) = 0 \]

Key Concepts

Definition of the Derivative

The cornerstone of differential calculus is the definition of the derivative. At its heart, the derivative provides a way to quantify how a function changes at any given point. It does this by taking the limit of the average rate of change of the function over an interval as that interval becomes infinitesimally small. This is mathematically expressed as:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]Here’s what each part means:

• \( f(x+h) \) and \( f(x) \): These represent the function values at \( x+h \) and \( x \) respectively.
• \( h \): A small increment, which approaches zero, representing the interval over which we measure the rate of change.
• \( \lim_{h \to 0} \): This limit ensures that we find the exact rate of change at the point \( x \) as \( h \) shrinks to nothing.

The result, \( f'(x) \), tells us the instantaneous rate of change, which is essentially the slope of the tangent line at the point \( x \) on the graph of the function.

Constant Function Derivative

A constant function is a function that always returns the same value, regardless of the input. Mathematically, a constant function can be expressed as \( f(x) = c \), where \( c \) is a constant. Now, let's explore its derivative.Substituting this into the derivative formula gives:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{c - c}{h} \]In this case, regardless of the input, the difference \( f(x+h) - f(x) \) becomes 0 because the function doesn’t change. Therefore, the whole fraction becomes:\[ \frac{0}{h} \]And therefore, the limit results in zero:\[ \lim_{h \to 0} 0 = 0 \]Hence, the derivative of a constant function is always zero. This result intuitively makes sense because a constant function doesn’t change; thus, it has no rate of change, or slope.

Instantaneous Rate of Change

The notion of the instantaneous rate of change serves as a foundation for understanding derivatives and insights into the behavior of functions. While the average rate of change calculates how much a function changes over an interval, the instantaneous rate gives us the rate at a precise point.This is formally represented by the derivative, \( f'(x) \). Consider it analogous to calculating the speed of a car at a specific moment in time rather than the average speed over a trip.Here’s why this concept is crucial:

• Precise Calculation: Unlike the average rate, which gives broad information over an interval, the instantaneous rate offers granularity at a specific instant.
• Real-Time Analysis: It's invaluable for real-time adjustments and predictions in various fields like physics, engineering, and economics, where understanding immediate changes is critical.
• Geometrical Interpretation: The slope of the tangent to the curve at a point reflects this rate of change, providing insights into the steepness or flatness of function graphs.

The derivative is a mathematical tool that turns the abstract idea of instantaneous change into a precise computation, capturing the dynamic nature of mathematical functions.