Problem 33 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The derivative \( f^{\prime}(x) = 0 \) for the constant function \( f(x) = 4 \).

Step 1: Recognize the Definition of the Derivative

The derivative of a function \( f(x) \) at a point \( x \) is defined by the limit \( f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). This is often called the limit definition of the derivative.

Step 2: Substitute the Function into the Derivative Definition

Given \( f(x) = 4 \), substitute this into the limit definition:\[f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{4 - 4}{h}\]

Step 3: Simplify the Expression

Calculate \( f(x+h) - f(x) \) for the constant function:\[4 - 4 = 0.\]Thus, the expression simplifies to:\[f^{\prime}(x) = \lim_{h \to 0} \frac{0}{h}\]

Step 4: Evaluate the Limit

As \( h \to 0 \), the expression \( \frac{0}{h} \) is zero for all \( h eq 0 \). Therefore, the limit is:\[f^{\prime}(x) = 0\] for any value of \( x \).

Key Concepts

Limit Definition of the Derivative

The limit definition of the derivative is a fundamental concept in calculus. It provides us with a way to calculate the instantaneous rate of change of a function at a specific point. Mathematically, it is expressed as:\[ f^{\prime}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]This formula helps us understand how the function behaves as its input values change infinitely close to a specific point. It essentially measures how steep the tangent line is to the graph of the function at that point.

• \( f(x+h) \) represents the function value slightly offset from \( x \).
• \( f(x) \) is the original function value at \( x \).
• \( h \) is the small change in \( x \).

By letting \( h \) approach zero, we find the exact slope of the tangent line at that point. This limit process is crucial because it moves us from average rates of change (over intervals) to instantaneous rates of change (at a point).

Constant Function

A constant function is one of the simplest types of functions in mathematics. For a constant function, the output value is the same for any input value. In formula terms, if \( f(x) = c \), where \( c \) is a constant, the graph forms a straight horizontal line across the y-axis. This means the slope of this line is zero.

So, when we use the limit definition of the derivative on a constant function, such as \( f(x) = 4 \),

• \( f(x+h) = 4 \)
• \( f(x) = 4 \)
• The subtraction \( f(x+h) - f(x) \) results in \( 0 \).

In consequence, the derivative of a constant function is always \( 0 \). This is because there is no change happening in the function's value, hence it doesn't vary as \( x \) changes.

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's a powerful tool for analyzing how things change and has wide applications in science and engineering.
One of its key components is differentiation, which was illustrated in this specific problem by finding the derivative of a constant function. Differentiation involves finding the derivative using the limit definition, as explored earlier, which measures the rate at which a function's value changes.

Calculus allows us to:

• Understand and predict system behavior by investigating rates of change (e.g., speed, growth rates).
• Analyze functional relationships to see how input variables affect outputs in dynamic systems.
• Solve physical problems involving motions, areas, and volumes through derivatives and integrals.

It opens the door to a deeper understanding of the physical world, providing techniques to solve complex problems that are difficult to address with algebra alone.