Problem 13 - Calculus with Concepts in Calculus

Question

Find all critical numbers of the given function. $$ f(z)=|z-2| $$

Step-by-Step Solution

Answer

The critical number of \( f(z) = |z-2| \) is \( z = 2 \).

Step 1: Understand Critical Numbers

Critical numbers of a function occur where the derivative is zero or undefined. Here, we must find the points where the derivative of \( f(z) = |z-2| \) is zero or where the function is not differentiable.

Step 2: Write the Absolute Value Function

The function \( f(z) = |z-2| \) can be expressed as a piecewise function: \( f(z) = z-2 \) if \( z \geq 2 \) and \( f(z) = -(z-2) \) if \( z < 2 \). This lets us handle the derivative in separate intervals.

Step 3: Differentiate Piecewise Function

Find the derivative for each piece of the piecewise function. For \( z \geq 2 \), \( f'(z) = 1 \). For \( z < 2 \), \( f'(z) = -1 \).

Step 4: Evaluate Where Derivative is Zero or Undefined

Since the derivatives, \( f'(z) = 1 \) and \( f'(z) = -1 \), are constants and never zero, the derivative cannot be zero at any point. However, the function \( f(z) = |z-2| \) is not differentiable where \( z = 2 \) due to the change in definition of the piecewise function.

Step 5: Identify Critical Numbers

The sole critical number is at the point where the derivative does not exist, which is \( z = 2 \).

Key Concepts

Differentiability

When delving into calculus, particularly the topic of critical numbers, understanding differentiability becomes pivotal. Differentiability of a function at a point means that the derivative exists at that point. If a function is differentiable, it implies smoothness at that point without any sharp turns or corners.

If you're checking for differentiability, look for:

• Continuity: Ensure the function is continuous around the point.
• Existence of the limit: The derivative must exist approaching the point from either side.

For the function \( f(z) = |z-2| \), differentiability is questionable at \( z = 2 \) because of the abrupt change in direction, making it non-differentiable at this particular point. This is where critical points often crop up, where the derivative either doesn't exist or changes abruptly.

Piecewise Function

A piecewise function is essentially a combination of various distinct functions, each applying to a specific part of the domain. This type of function often arises in real-world situations like taxes or shipping rates, creating different rules for different scenarios.

To better understand, consider the function \( f(z) = |z-2| \). We can express it as

• \( f(z) = z-2 \) for \( z \geq 2 \)
• \( f(z) = -(z-2) \) for \( z < 2 \)

This piecewise definition helps in calculating derivatives for each part of the function individually, as each 'piece' can be treated like its own unique function. Such definitions are crucial, especially when identifying potential critical points, as they let you isolate intervals to examine differentiability.

Derivative

The derivative is a central concept in calculus, representing the rate of change of a function. It tells us how a function behaves as our input changes. In graphical terms, it's the slope of the tangent line at a point on the function's curve.

To find the derivative of a piecewise function like \( f(z) = |z-2| \), we examine each interval separately:

• For \( z \, \geq \, 2 \), \( f'(z) = 1 \)
• For \( z \, < \, 2 \), \( f'(z) = -1 \)

These derivatives are constant, indicating linear behavior within each interval. However, at \( z = 2 \), the derivative does not exist due to a sharp point in the function, making this a critical number. Recognizing where the derivative is zero or does not exist helps us find crucial points on the function that might represent peaks, troughs or sudden changes.