Problem 61 - University Calculus: Early Transcendentals

Question

Graph the derivative of \(f(x)=|x|\) Then graph \(y=(|x|-0) /(x-0)=|x| / x .\) What can you conclude?

Step-by-Step Solution

Answer

Both graphs represent a step function with values 1 for x > 0 and -1 for x < 0, with a discontinuity at x = 0.

Step 1: Understand the Function

The function given is \( f(x) = |x| \), which is the absolute value function. It produces a 'V' shape, where for \( x < 0 \), \( f(x) = -x \), and for \( x \geq 0 \), \( f(x) = x \).

Step 2: Analyze the Derivative

The derivative of \( f(x) = |x| \) depends on its piecewise definition:- For \( x > 0 \), the derivative \( f'(x) = 1 \), because \( f(x) = x \) and the derivative of \( x \) is 1.- For \( x < 0 \), the derivative \( f'(x) = -1 \), because \( f(x) = -x \) and the derivative of \(-x\) is -1.- At \( x = 0 \), the derivative is undefined because there is a sharp corner at the origin where the left- and right-hand limits of the derivative do not agree.

Step 3: Graph the Derivative

Graph the function where:- For \( x > 0 \), draw a horizontal line at \( y = 1 \).- For \( x < 0 \), draw a horizontal line at \( y = -1 \).- There is an open circle at \( x = 0 \) since the derivative is undefined there.

Step 4: Understand the New Expression

The new expression is \( y = \frac{|x|}{x} \). This function returns the sign of \( x \):- For \( x > 0 \), \( y = 1 \).- For \( x < 0 \), \( y = -1 \).- At \( x = 0 \), the expression is undefined since division by zero is not possible.

Step 5: Graph the New Function

Graph \( y = \frac{|x|}{x} \) which results in:- A horizontal line at \( y = 1 \) for \( x > 0 \).- A horizontal line at \( y = -1 \) for \( x < 0 \).- An open circle at \( x = 0 \) because the function is undefined there.

Step 6: Conclusion

The graph of the derivative of \( f(x) = |x| \) and the graph of \( y = \frac{|x|}{x} \) are the same. Both are represented by a step function with discontinuity at \( x = 0 \). This shows the derivative reflects the rate of change for the absolute value function in separate positive and negative regions.

Key Concepts

Derivative of Absolute Value

The absolute value function, represented as \( f(x) = |x| \), is a fundamental concept in mathematics. This function forms a 'V' shape graph, where the values are always non-negative. But when computing its derivative, we encounter some unique characteristics due to its sharp corner at \( x = 0 \).

When deriving step by step:

• For \( x > 0 \), \( f(x) = x \) and thus, the derivative is \( f'(x) = 1 \). This means the slope of the graph is a constant 1 on the positive side.
• For \( x < 0 \), \( f(x) = -x \), leading to a derivative of \( f'(x) = -1 \). This shows the slope here is a constant -1 on the negative side.
• At \( x = 0 \), the derivative is undefined. This happens because the absolute value function has a sharp turn — known as a cusp — at this point, and the left-hand and right-hand slopes don’t match up.

This derivative behavior is quite typical for functions with corners or cusps. It highlights how derivatives reflect the instantaneous rate of change and show discontinuities at such sharp transitions.

Step Function

The derivative of the absolute value function leads us to a step function. Step functions have a set of constant values over their intervals.

In the case of the derivative of \( f(x) = |x| \):

• For \( x > 0 \), the value of the function is constantly \( 1 \).
• For \( x < 0 \), it is constantly \( -1 \).
• At \( x = 0 \), the function is undefined, creating a break or jump at this point.

The visual representation of this step function shows two horizontal lines: one at \( y=1 \) and one at \( y=-1 \), with a discontinuity right at \( x=0 \), marked by an open circle.

Step functions are useful in modeling scenarios with sudden changes, because they capture the abrupt shifts between different constant rates, exactly as seen with the derivative of the absolute value function.

Piecewise Function

Piecewise functions are an essential mathematical tool, where you define a function using different expressions based on different intervals of the independent variable, \( x \). The absolute value function itself is a good example of a piecewise function.

For the formula \( f(x) = |x| \):

• We have \( f(x) = x \) when \( x \geq 0 \).
• For \( x < 0 \), \( f(x) = -x \).

Each part of the function applies to a specific section of the \( x \)-axis, creating a clear transition at \( x = 0 \). This characteristic helps manage and express mathematical functions that behave differently over distinct intervals.

The piecewise nature of the absolute value function is what leads to its derivative being undefined at \( x = 0 \), as the separate behaviors of \( x \) greater than or less than zero don't seamlessly connect. Understanding piecewise functions allows students to gain insight into evaluating and handling more complex mathematical expressions.