Problem 56 - Calculus

Question

Draw the graph of a continuous function \(y=f(x)\) that satisfies the following three conditions: \(\cdot f^{\prime}(x)>0\) for \(x<-2\) \(\cdot f^{\prime}(x)<0\) for \(-22\)

Step-by-Step Solution

Answer

Draw an increasing curve for \(x<-2\), a decreasing curve for \(-2 < x < 2\), and a flat line for \(x > 2\).

Step 1: Analyze Condition 1

The given condition is \( f'(x) > 0 \) for \( x < -2 \). This means the function is increasing in this interval. As \( x \) approaches \(-2\) from the left, the slope is positive, so we need to draw a rising curve for \( x < -2 \).

Step 2: Analyze Condition 2

The second condition is \( f'(x) < 0 \) for \( -2 < x < 2 \). This indicates the function is decreasing in this interval. Therefore, from \( x = -2 \) to \( x = 2 \), we draw a falling curve, ensuring it starts from a high point and moves downward.

Step 3: Analyze Condition 3

The third condition is \( f'(x) = 0 \) for \( x > 2 \). This signifies the function has a constant slope of zero, indicating a flat or horizontal line. Therefore, for \( x > 2 \), the function should be a horizontal line.

Step 4: Combine the Conditions

To represent all three conditions accurately, we begin the curve with an increasing segment for \( x < -2 \), followed by a decreasing segment for \( -2 < x < 2 \), and finally, a flat horizontal line for \( x > 2 \). Ensure the transitions at \( x = -2 \) and \( x = 2 \) are smooth, as the function is continuous.

Step 5: Graph the Function

Draw the increasing section that rises as \( x \) approaches \(-2\) from the left. At \( x = -2 \), smoothly transition to a decreasing segment until \( x = 2 \). Finally, continue with a horizontal line starting at \( x = 2 \) to the right, indicating that the derivative is zero in this region.

Key Concepts

Graph Analysis

The first step in understanding this problem involves analyzing the graph of the continuous function. When analyzing a graph, it's crucial to consider how the function behaves over different intervals, as indicated by the conditions.
Identifying where the function rises or falls is essential:

• For the interval where the derivative is greater than zero (\( f'(x) > 0 \), the graph will incline upwards, indicating an increasing function.
• In the section where;\( f'(x) < 0 \), the graph descends, symbolizing a decreasing trend.
• Finally, when the derivative equals zero (\( f'(x) = 0 \), it results in a flat, horizontal line.

The transitions between the increasing, decreasing, and flat segments must be smooth to ensure the function remains continuous. Envisioning these features along a plotted graph can help visualize the overall shape of the function and the changes in direction as you cross key points like \( x = -2 \) and \( x = 2 \).

Derivative Rules

Understanding derivative rules is central to analyzing the changes in a function. Derivatives tell us the slope of a function at any given point, defining whether it's rising, falling, or remaining constant.
For this function, the rules of derivatives apply to determine its behavior on different intervals:

• The condition \( f'(x) > 0 \) implies that the slope is positive, ensuring the function is increasing.
• A condition showing \( f'(x) < 0 \) implies a negative slope, marking a decreasing function.
• When\( f'(x) = 0 \), the slope is zero, resulting in a flat line.

It is important to apply these rules accurately to predict how the function behaves. This understanding provides deeper insight into how graphs change shape across different intervals, as well as the reasons behind rising or falling sections of the graph, directly stemming from these derivative conditions.

Slope Interpretation

Slope interpretation refers to understanding the slopes of the function across its domain. In calculus, slope directly relates to the first derivative of a function.
By interpreting the slope, students can understand:

• An increasing function when slope \( > 0 \)
• A decreasing function when slope \( < 0 \)
• A constant function when slope \( = 0 \)

The slope between points implies movement direction on the graph.
For classroom purposes, relating these slopes to visual depictions on a graph can provide students an intuitive grasp of the function’s dynamic changes.
This understanding is crucial in transitions between positive, negative, or zero slope areas like points \( x = -2 \) and \( x = 2 \). By visualizing these slopes, the concept of continuity becomes more tangible. The continuous nature of these transitions helps avoid abrupt changes, making the graph smooth and cohesive.