Problem 28 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. If the tangent line to the graph of \(y=f(x)\) at \(x=-2\) has negative slope, then \(f^{\prime}(-2)<0\)

Step-by-Step Solution

Answer

True, because a negative slope of the tangent line means the derivative is negative.

Step 1: Understanding the Problem

We are asked to assess whether the statement about the derivative at a specific point on the graph of a function is true or false. Specifically, we need to investigate what is implied by a tangent line having a negative slope at a point.

Step 2: Concept of Tangent Line Slope

The slope of the tangent line to the graph of a function at any given point is represented by the derivative of the function at that point. If the derivative is negative, the slope of the tangent line is negative, indicating the function is decreasing at that point.

Step 3: Applying the Description

According to the statement, the tangent line to the graph of \(y=f(x)\) at \(x=-2\) has a negative slope. Therefore, from our understanding, this implies that \(f^{\prime}(-2)<0\), because the derivative represents the slope of the tangent line.

Step 4: Conclusion Based on Analysis

Given that having a negative slope for the tangent line directly translates to a negative derivative, we can confirm that the statement is true.

Key Concepts

Tangent Line

In calculus, the tangent line at a given point on a curve is a straight line that just "touches" the curve at that point. The most important property of a tangent line is its slope, which provides us with insight into the behavior of the function at that particular point.

• The tangent line is locally straight and provides the best linear approximation of the function near that point.
• Even if a curve can be complex, the tangent line helps us understand how the curve behaves under a microscope at the specific point of tangency.
• This slope of the tangent line is determined by the derivative, which we calculate at the point of interest.

Thus, when we're told the tangent line to a function at a specific point, like at \(x=-2\), has a negative slope, it means that the function's rate of change is negative at that point.

Slope of a Curve

The slope of a curve at a specific point provides critical information about the curve's behavior at that point. The slope tells us:

• Whether the function is increasing or decreasing at that point.
• The steepness of the curve, meaning how sharply it rises or falls.

In calculus, the slope of a curve at a point is mathematically expressed as the derivative of the function at that point. For instance, given a function \(y=f(x)\), the slope of the curve at \(x=-2\) is expressed as \(f'(x)\) evaluated at \(-2\), or \(f'(-2)\). If this derivative is less than zero, \(f'(-2) < 0\), it implies:

• The slope of the tangent line is negative, indicating the function is moving downward as \(x\) increases past \(-2\).
• The function is decreasing at that point, showing a downward trend.

Understanding the slope of a curve can therefore help us draw conclusions about the behavior of the function at any point.

Function Analysis

Function analysis is the process of examining the characteristics and behaviors of functions to understand their properties and predict their trends. By analyzing a function, you can:

• Determine critical points where the function changes its direction.
• Understand where the function is increasing or decreasing.
• Find points where the function reaches a local maximum or minimum.

The derivative plays a major role in function analysis, as it essentially provides the rate of change of the function. By examining the sign of the derivative:

• Positive derivative values indicate increasing function behavior.
• Negative derivative values indicate decreasing function behavior.

When analyzing the function \(y=f(x)\) around \(x=-2\), for example, figuring out that \(f'(-2)<0\) helps us conclude the function is decreasing at that point. This kind of analysis empowers us to predict and understand the behavior and trend of the function on a wider scale, whether it describes real-world phenomena or abstract mathematical concepts.