Problem 28 - Calculus - AP Edition

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; the statement is correct because the derivative being less than zero indicates a negative slope.

Step 1: Understand the Statement Given

We are asked to determine the validity of the statement: whether the slope of the tangent line to the graph of a function at a specific point is related to the derivative of the function at that point being negative.

Step 2: Recall Key Concepts

The slope of the tangent line to the curve defined by the function \(y = f(x)\) at any point \(x = a\) is given by the derivative \(f^{\prime}(a)\). If \(f^{\prime}(a)<0\), it means the slope of the tangent line at that point is negative, indicating that the function is decreasing at \(x=a\).

Step 3: Analyze the Condition

The tangent line to the graph of \(y = f(x)\) at \(x = -2\) has a negative slope, meaning \(f^{\prime}(-2)<0\). By definition, a negative slope corresponds to a derivative that is less than zero.

Step 4: Formulate Conclusion

Since the definition of a derivative as the slope of the tangent line directly connects to having a negative derivative when the slope is negative, the statement provided is true.

Key Concepts

Tangent Line

In calculus, a tangent line to a curve at a given point provides the best linear approximation of the curve near that point. Simply put, it is a straight line that just grazes the curve at that single point and has the same slope as the curve does at that exact location. Mathematically, for a function \( y = f(x) \), the tangent line at a point \( x = a \) can be found using the derivative of the function, \( f'(a) \), which determines the slope. This is why the tangent line is often said to 'touch' the curve at one point without cutting through it.

• Key feature: The tangent line represents the instantaneous rate of change of the function at a particular point.
• Importance: It helps in understanding how a function behaves in a very small neighborhood around a given point.

Understanding the tangent line is essential for visualizing and interpreting the behavior of functions in calculus.

Slope of a Function

The slope of a function at a particular point is determined by the derivative at that point. It indicates how steeply the graph of the function is rising or falling at that point. For any curve defined by a function \( y = f(x) \), the slope at a particular point \( x = a \) is given by the derivative \( f'(a) \). If \( f'(a) > 0 \), the function is increasing at \( x = a \); if \( f'(a) < 0 \), the function is decreasing. The bigger the absolute value of the derivative, the steeper the slope.

• Positive slope: Indicates the function is increasing as you move from left to right.
• Negative slope: Indicates the function is decreasing.
• Zero slope: Indicates a horizontal tangent line, often occurring at the peak, trough, or point of inflection of the function.

Knowing the slope of a function at any given point helps in predicting the behavior of the function in that region.

Function Decreasing

When a function is decreasing on a certain interval, it means that as you move from left to right along the graph, the function's values drop. This occurs when the slope of the tangent line is negative, or equivalently, when the derivative of the function \( f'(x) \) is negative within that interval. A negative derivative indicates that the function outputs smaller and smaller values as the input \( x \) increases, thus reflecting a downward trend.

• Example: Consider the function \( f(x) = -2x + 3 \). Its derivative is constant at \( -2 \); hence, it is always decreasing, reflecting a straight line with a negative slope.
• Visualizing decrease: A downward slope on the graph signifies that for every unit increase along the x-axis, the y-value (or output) of the function declines.

Recognizing when a function is decreasing is crucial for understanding its overall trend and behavior across specified intervals.