Problem 68 - Precalculus : Building Concepts and Connections

Question

Suppose \(f\) is constant on an interval \([a, b] .\) Show that the average rate of change of \(f\) on \([a, b]\) is zero.

Step-by-Step Solution

Answer

The average rate of change of a constant function \(f\) on an interval [a, b] is zero.

Step 1: Define the Average Rate of Change

The average rate of change of a function \(f\) on an interval [a, b] is given by the formula: \[Average Rate of Change = \frac{f(b) - f(a)}{b-a}\]

Step 2: Apply the Function Definition

Since \(f\) is a constant function, we know that \(f(b) = f(a)\) because the value of function \(f\) remains constant for all values of \(x\) in the interval [a, b] including \(a\) and \(b\). So substitute \(f(b) = f(a)\) into the formula from Step 1 which simplifies the equation to: \[Average Rate of Change = \frac{f(a) - f(a)}{b-a}\]

Step 3: Simplify the Equation

\(f(a) - f(a)\) is equal to zero because subtracting the same number from itself always yields zero. The equation then simplifies to \[Average Rate of Change = \frac{0}{b-a}\] Regardless of the values of \(b\) and \(a\), the numerator is zero so the fraction is zero. Therefore the average rate of change of \(f\) on [a, b] is zero.

Key Concepts

Constant Function

A constant function is a simple, yet powerful, type of mathematical function. It is defined by a rule that assigns the same value to every input in its domain. This means that no matter what value of "x" you choose, the output remains constant at some fixed number "c". Mathematically, we can express a constant function as:

• \(f(x) = c\)

where \(c\) is a constant real number.
Constant functions appear as horizontal lines on a graph, showing no increase or decrease.
When dealing with a constant function, one of its essential properties is that its average rate of change over any interval, say \([a, b]\), is always zero. Since a constant function does not vary, both \(f(a)\) and \(f(b)\) are equal, leading to a zero difference in its outputs, as explained in the problem's solution.

Interval Analysis

Interval analysis involves examining the behavior of functions over specific spans of their domains, called intervals. An interval could be made up of an isolated range, like \([a, b]\), or it can extend over infinite regions.
For constant functions, interval analysis helps confirm that their behavior is simple and predictable across any interval.

• Every point in the interval \([a, b]\) will return the same function value, \(c\).
• This makes calculations straightforward, such as determining the average rate of change, which will always be zero for constant functions.

Understanding how functions behave over intervals can greatly assist in visualizing concepts like continuity or constancy, ensuring a clear grasp of their fundamental characteristics.

Function Properties

The properties of a function describe how it behaves and interacts within its defined domain. A constant function is notably simple, with defining characteristics that make it unique compared to other functions.
Key properties include:

• Domain: All real numbers. A constant function accepts any real number as an input.
• Range: The single constant value. The output remains the same regardless of the input.
• Derivative: The derivative of a constant function is zero, establishing its unchanging nature.

Understanding function properties is crucial, as they determine what kind of changes occur in a function across its interval, like linear growth for linear functions or no change for constant ones. A constant function, by definition, embodies uniformity in output, explaining why its average rate of change across any interval, such as \([a, b]\), is zero.