Problem 37 - University Calculus: Early Transcendentals

Question

Does the graph of $$f(x)=\left\\{\begin{aligned}-1, & x<0 \\\0, & x=0 \\\1, & x>0\end{aligned}\right.$$ have a vertical tangent at the origin? Give reasons for your answer.

Step-by-Step Solution

Answer

No, the graph does not have a vertical tangent at the origin because the derivative is zero on both sides and doesn't exist at the origin.

Step 1: Understanding the Problem

We need to determine if the piecewise function \(f(x)\) has a vertical tangent at the origin \((0,0)\). A vertical tangent occurs when the derivative approaches infinity or negative infinity at that point.

Step 2: Analyzing the Function

The given function \( f(x) \) is defined as a piecewise function: \(-1\) for \( x < 0 \), \(0\) for \( x = 0 \), and \(1\) for \( x > 0 \). The function makes a jump at \( x = 0 \).

Step 3: Considering Derivatives

Since the function has a different constant value on either side of zero (-1 for \( x < 0 \) and 1 for \( x > 0 \)), its derivative does not exist at \( x = 0 \). For \( x < 0 \) and \( x > 0 \), the derivative of a constant is zero.

Step 4: Conclusion

Because the derivative of the function is zero for \( x < 0 \) and \( x > 0 \) and the function is not differentiable at \( x = 0 \) due to the jump from -1 to 1, there is no vertical tangent at the origin.

Key Concepts

Piecewise Function

A piecewise function is a type of function that has different expressions or "pieces" for different intervals of its domain. This means that the rule for computing the function value changes depending on the input. For the function given in the exercise:

• When the input \( x \) is less than 0, the function value is -1.
• At exactly 0, the function value is 0.
• For anything greater than 0, it changes to 1.

Because of these abrupt changes or "jumps" in the function values at specific points (in this case, at the origin), analyzing limits or derivatives can become more complex. It's crucial to understand each piece separately to determine behavior, such as continuity or differentiability, at those transition points.

Vertical Tangent

A vertical tangent is a special type of tangent line to a curve at a point where the curve becomes vertical. In mathematical terms, this occurs when the slope of the tangent line approaches infinity or negative infinity at that point. For finding a vertical tangent in a function:

• The first step is to compute the derivative of the function.
• Check for points where the derivative doesn't exist or tends towards infinity.

In the context of a piecewise function, this involves analyzing the derivative of each piece and the behavior at the boundaries. However, the given piecewise function in the original exercise does not possess a vertical tangent at the origin because there's a discontinuity at that point, making the derivative undefined. Thus, when looking for vertical tangents, understanding both the derivative and function continuity is key.

Derivative

A derivative represents the rate of change or the slope of a function at a specific point. It provides valuable information about the function's behavior and is fundamental in calculus.For piecewise functions, derivatives need special attention because the function is defined by different rules in different intervals. Here are key points to remember:

• The derivative of a constant function is zero.
• At points of discontinuity, like with the given exercise, the derivative may not exist.
• Analyzing each specific piece and its limits can reveal important behavior trends.

In the given exercise, the derivative is zero for both pieces where \( x < 0 \) and \( x > 0 \), since they are constant values. At \( x = 0 \), there's a jump, thus the derivative doesn't exist, confirming the absence of a vertical tangent at that point.