Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

A slope field is given for a differential equation of the form \(y^{\prime}=f(x, y) .\) Use the slope field to sketch the solution that satisfies the given initial condition. In each case, find \(\lim _{x \rightarrow \infty} y(x)\) and approximate \(y(2) .\) $$ y(1)=3 $$

Step-by-Step Solution

Answer

Approximate \( y(2) \) from the sketch; determine the limit \( \lim_{x \rightarrow \infty} y(x) \) from the curve's behavior.

Step 1: Understand the Slope Field

A slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation \( y' = f(x, y) \). Each small line segment or slope in the field indicates the slope of the solution curve at that point. Begin by examining the slope field to understand the general behavior of potential solution curves.

Step 2: Identify the Initial Condition

The given initial condition is \( y(1) = 3 \). Identify the point \((1, 3)\) on the slope field. The solution curve must pass through this point because it represents the initial value of the function \(y(x)\).

Step 3: Sketch the Solution Curve

Starting from the initial point \((1, 3)\), sketch the solution that follows the direction (or slope) indicated by the slope field. The sketched curve should smoothly follow the direction of the small lines, representing the behavior of the differential equation solution.

Step 4: Estimate \( y(2) \)

From the sketched solution curve, locate \( x = 2 \) on the x-axis. Approximate \( y(2) \) by determining the value of \( y \) at this \( x \) value based on the path of the sketched curve. This will involve looking at the vertical axis and estimating where the curve is positioned above \( x = 2 \).

Step 5: Determine \( \lim_{x \to \infty} y(x) \)

Observe the general trend of the sketched solution curve as \( x \) increases. Consider if the curve approaches a horizontal asymptote or continues to increase/decrease without bound. Based on this observation, determine \( \lim_{x \to \infty} y(x) \).

Key Concepts

Understanding Slope Fields

A slope field is like a map showing the direction of solutions for a differential equation without solving it explicitly. Imagine it as a scattered collection of tiny arrows or lines that illustrate the slope at various points in the plane.

• These lines represent the derivative, or the rate of change, of the function at those specific points.
• For the differential equation given (\(y' = f(x, y) \)), the slope field helps us visualize potential solutions.

By examining the slope field, you can anticipate how a solution curve might behave. It provides a snapshot of the direction solutions are likely to take at every point, allowing us to sketch out solution paths with greater ease.

Role of Initial Condition

The initial condition is a crucial piece of information that specifies the starting point for our solution curve on a slope field. It tells us exactly where a particular solution should begin.

• In our example, the initial condition given is \(y(1) = 3\). This corresponds to the point \((1, 3)\) on the graph.
• This starting point ensures that the solution curve we sketch will be unique because it must pass through this specified spot on the slope field.

With the initial condition, we eliminate the ambiguity of having multiple potential curves by selecting the one that perfectly fits through the initial point \((1, 3)\). This provides a solid grounding for accurately predicting the behavior of our solution curve.

Sketching the Solution Curve

Once we've pinpointed the initial condition, sketching the solution curve involves following the flow of the slope field from the initial point.

• The sketched curve should smoothly follow the direction indicated by the tiny lines of the slope field.
• As you draw the curve, its path should trace these direction lines, reflecting the unique solution to the differential equation through the given initial point.

After sketching, estimating values like \(y(2)\) involves locating where the sketched curve intersects the x-value at \(x = 2\). From here, assess the position along the y-axis to approximate this value. This process highlights how the visual guides of a slope field can help us derive solutions step by step.

Importance of Asymptotes in Solutions

In the study of solution curves, asymptotes play a key role in understanding their long-term behavior, especially when examining the limits as \(x\) approaches infinity.

• An asymptote is a line that a curve approaches as \(x\) continues to stretch towards a very large value or changes drastically.
• For our solution curve, observing the general trend as it moves rightward on the x-axis will help determine if it levels off to a particular value or continues to rise or fall.

When we analyze the behavior of the curve for \(\lim_{x \to \infty} y(x)\), we seek to find if it approaches a horizontal asymptote, signifying stabilization, or if it diverges, indicating boundless increase or decrease. This information provides deep insights into the nature of the differential equation's solutions.