Problem 10 - Calculus

Question

Sketch a slope field with the following properties. (Draw at least ten line segments, including some with \(x<0\) with \(x>0, \text { and with } x=0 .)\) $$\begin{aligned} &\frac{d y}{d x}>0 \text { for } x<0,\\\ &\frac{d y}{d x}<0 \text { for } x>0,\\\ &\frac{d y}{d x}=0 \text { for } x=0 \end{aligned}$$

Step-by-Step Solution

Answer

Draw upward segments for \(x<0\), downward for \(x>0\), and horizontal for \(x=0\).

Step 1: Understand the Given Conditions

The exercise requires sketching a slope field based on a set of given properties for the derivative \( \frac{dy}{dx} \). The derivative is positive for \( x < 0 \), negative for \( x > 0 \), and zero for \( x = 0 \). This indicates that the line segments at any point \((x, y)\) need to reflect these slope values.

Step 2: Determine Slope Directions for x < 0

Since \( \frac{d y}{d x} > 0 \) for \( x < 0 \), the slopes of the line segments should be positive. This means that the line segments will be angled upwards as they move from left to right when \( x < 0 \).

Step 3: Determine Slope Directions for x > 0

For \( x > 0 \), the derivative \( \frac{d y}{d x} < 0 \), indicating that the line segments should be negative. This results in the line segments being angled downwards as they move from left to right for \( x > 0 \).

Step 4: Determine Slope at x = 0

Since \( \frac{d y}{d x} = 0 \) for \( x = 0 \), the slopes of the line segments will be horizontal. This means the line segments will not be angled either upwards or downwards when \( x = 0 \).

Step 5: Sketch the Slope Field

Start by drawing a Cartesian plane. For \( x < 0 \), sketch at least five line segments oriented upwards. For \( x > 0 \), draw at least five line segments oriented downwards. At \( x = 0 \), draw line segments that are horizontal. Ensure a variety of \( y \)-values to complete the slope field.

Key Concepts

Derivative

In mathematics, the derivative of a function measures how the function's output value changes with respect to a change in the input value. It is a fundamental concept in calculus used to find the rate at which a quantity changes. In mathematical terms, if you have a function \( f(x) \), the derivative is often written as \( \frac{df}{dx} \) or \( f'(x) \).

The derivative is crucial for analyzing the properties of graphs and understanding slope fields. In a slope field, the derivative indicates the direction and steepness of line segments at various points in the plane.

• A positive derivative means the function is increasing at that point.
• A negative derivative indicates the function is decreasing.
• A zero derivative signifies a constant function at that point.

The concept of the derivative allows us to create slope fields that visually represent these changes in behavior.

Positive Slope

A positive slope in a slope field indicates that the line segments are rising as you move from left to right. When dealing with the derivative \( \frac{dy}{dx} > 0 \), the slope of the line segments will be positive. This means that as you move from one point to another on the graph, the line segments will have an upward trend.

Here's what a positive slope tells us in more detail:

• A positive slope suggests that the function is increasing, reflecting a rise in the value of \( y \) as \( x \) increases.
• On the graph, this results in line segments that slant upwards, forming an angle with the horizontal axis.

In our original exercise, for any point where \( x < 0 \), the line segments in the slope field must depict an upward motion, showing the positive nature of the derivative.

Negative Slope

When a slope is negative, it means that the line segments in a slope field will fall as you move from left to right. With \( \frac{dy}{dx} < 0 \), the slope of the line segments is negative, indicating a downward trend.

Understanding the negative slope involves:

• A negative slope indicates that the function is decreasing, meaning the \( y \)-value drops as \( x \) increases.
• Graphically, this is seen as line segments slanting downwards, forming an angle with the horizontal in the opposite direction of a positive slope.

In the context of the given problem, when \( x > 0 \), the slope field should contain these downward-sloping segments, representing the negative derivative.

Horizontal Slope

A horizontal slope refers to a scenario where \( \frac{dy}{dx} = 0 \). In this case, the slope of the line segments is neither rising nor falling, which results in horizontal lines.

This particular condition provides:

• A horizontal slope indicates that the function's value does not change with a change in \( x \), showing a constant function.
• This is visually represented as a flat, horizontal line in the slope field where all the line segments form straight lines parallel to the \( x \)-axis.

Specifically, in the exercise, at \( x = 0 \), the slope must be horizontal, showing that there is no change in the function's value at those points.