Problem 29 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If the graph of \(f^{\prime}\) has a vertical asymptote at \(x=1,\) then \(f\) cannot be continuous at \(x=1\)

Step-by-Step Solution

Answer

False; \(f\) can still be continuous despite \(f'\) having a vertical asymptote at \(x=1\).

Step 1: Understanding Vertical Asymptote

A vertical asymptote in the graph of a function indicates that the function approaches infinity or negative infinity as it gets close to a certain x-value. When this occurs at a point \(x = 1\) for \(f'\), it suggests that the derivative becomes undefined or infinte at this point.

Step 2: Relationship between Continuity and Differentiability

A function \(f\) is differentiable at a point \(x = 1\) only if it is continuous at \(x = 1\). If the derivative \(f'\) has a vertical asymptote at \(x = 1\), \(f\) is not differentiable at that point.

Step 3: Analyzing Continuity of f

While a vertical asymptote in \(f'\) indicates non-differentiability of \(f\), it does not necessarily imply that \(f\) is discontinuous. A function \(f\) can be continuous yet have points where it is not differentiable. Thus, the existence of a vertical asymptote in \(f'\) does not guarantee discontinuity in \(f\) itself.

Step 4: Conclusion: Determining the Statement's Validity

The statement "If the graph of \(f^{\prime}\) has a vertical asymptote at \(x=1,\) then \(f\) cannot be continuous at \(x=1\)" is false. The absence of differentiability due to the vertical asymptote in \(f'\) does not preclude the continuity of \(f\) at \(x = 1\).

Key Concepts

Vertical Asymptote

Vertical asymptotes are fascinating occurrences on function graphs. They represent places where the graph shoots up to infinity or plunges down to negative infinity as it nears a particular x-value. Whenever you stumble upon a vertical asymptote for the derivative of a function, such as at \( x = 1 \), it reveals that the slope of the tangent (i.e., the derivative) is undefined there. This provides important insights into the behavior of the function at that point.

Think of it as trying to balance a pencil upright on your finger. As the pencil tilts more and more, it reaches a critical point where it either falls forward or backward very quickly. That's somewhat akin to what happens mathematically with vertical asymptotes.

• A vertical asymptote at \( x = 1 \) means the derivative approaches infinity or negative infinity.
• The derivative \( f' \) is undefined at \( x = 1 \).
• This asymptote gives hints about the non-differentiability of the parent function \( f \).

Functions

Functions are the building blocks for many mathematical studies. A function maps every input to exactly one output, much like a vending machine where each button corresponds to a particular snack. When you're looking at questions related to continuity and differentiability, it can be helpful to distinguish how these terms relate to functions like \( f \) and their derivatives.

Continuity is one of the key features of functions. We say that a function \( f \) is continuous at a point \( x = 1 \) if you can draw the function at that point without lifting your pencil.

• Continuity ensures there are no jumps, breaks, or holes at \( x = 1 \).
• A continuous function allows consistent values when approaching from any direction.
• It is possible for \( f \) to be continuous even if \( f' \) has a vertical asymptote.

Derivatives

Derivatives are fundamental tools to understand the rate at which functions change. They give us the slope of the function's graph at any given point and are key in understanding the concept of differentiability.

If you encounter a vertical asymptote in the derivative \( f' \), it suggests rapid changes and that the derivative does not exist at that point. However, this does not automatically indicate that the original function \( f \) is discontinuous.

• A derivative helps determine a function's instantaneous rate of change.
• Nonexistence or infinite derivative suggests non-differentiability at that point.
• Continuity and differentiability are closely related, but a function can still be continuous even if not differentiable.

Understanding these concepts helps you grasp deeper insights into how even small changes in the derivative \( f' \) can impact, but not absolve, the continuity of \( f \).