Problem 47 - Calculus Early Transcendentals

Question

Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. $$y^{\prime}(x)=\sin x, y(-2)=2$$

Step-by-Step Solution

Answer

Answer: To sketch a solution curve using an initial condition and direction field, follow these steps: 1. Locate the initial condition point on the x-y plane and mark it with a small dot. 2. Observe the direction of arrows around the initial point. 3. Sketch the path that the solution curve would follow based on the direction of arrows, ensuring that the curve is tangent to the direction field arrows at all points. By following these steps, you will obtain a sketch of the solution curve that satisfies the given differential equation and initial condition.

Step 1: Sketch the direction field

To sketch the direction field for the given first-order differential equation, follow these steps: 1. Choose equally spaced points in the given window \([-2,2] \times [-2,2]\). 2. Compute the corresponding slope y'(x) at each point by plugging the x-value into the equation \(y^{\prime}(x)=\sin x\). 3. Represent the slope at each point by drawing a small line segment (called a "direction arrow") that has the calculated slope. Once you have placed these direction arrows throughout the x-y plane, you will have a visual representation of the equation's slope.

Step 2: Sketch the solution curve

Now that we have sketched the direction field, let us sketch the solution curve with an initial condition y(-2) = 2: 1. Locate the point (-2, 2) on the x-y plane and draw a small dot at that point to represent the initial condition. 2. Observe the direction of arrows around the initial point. 3. Sketch the path that the solution curve would follow based on the direction of arrows, ensuring that the curve is tangent to the direction field arrows at all points. By following these steps, you will arrive at a graph of the solution curve that satisfies the given differential equation and initial condition.

Key Concepts

First-Order Differential Equations

First-order differential equations form a fundamental part of mathematics and sciences. They can be summarized as equations involving the derivatives of a function. This means they describe the rate at which quantities change. A typical first-order differential equation can be expressed as \(y' = f(x, y)\), where \(y'\) represents the derivative of \(y\) with respect to \(x\), and \(f(x, y)\) is some given function involving \(x\) and \(y\). In the exercise, the specific equation given is \(y' = \sin x\). This simplifies things since it only depends on \(x\).

• The first-order differential equations are important in modeling real-world scenarios like population growth, chemical reactions, and heat conduction.
• The beauty of these types of equations is their versatility and relevance across different fields.
• Such equations often require an initial condition to find a unique solution, making them even more powerful.

Understanding that a differential equation describes how quantities change can help you see why direction fields and calculating slopes at various points become essential tools in solving or visualizing them.

Initial Condition

An initial condition in the context of a differential equation is a specific value or set of values provided to find a unique solution to the problem. An initial condition will typically specify the value of the function, such as \(y(x)\), at a particular point \(x = a\). In the given exercise, the initial condition is \(y(-2) = 2\).

• Initial conditions help to solve differential equations by anchoring the solution curve to a specific point in the graph.
• In practice, having an initial condition means that there is a definite starting point from where you begin tracing the solution curve.
• This initial point ensures that among many possible solutions, your answer is unique.

The initial condition essentially acts as a starting guide. From this point, you follow the direction indicated by the slope field to determine how \(y\) evolves with \(x\).
So, whenever you have a differential equation along with an initial condition, you can confidently trace the curve knowing that it will satisfy the mathematical requirements at that point.

Slope Field

A slope field, also known as a direction field, visually represents the solutions to a first-order differential equation without solving it analytically. It does this by displaying small line segments or arrows on the coordinate plane, each with a slope given by the differential equation at that point. In this exercise, the slope field relates to the equation \(y' = \sin x\).

• Each arrow on the slope field represents the slope of the solution at that point, helping to visualize how the solution behaves.
• The arrows guide the sketching of the solution curve by providing a visual path that the curve must follow.
• Slope fields are particularly useful when analytic solutions are difficult or impossible to obtain.

To create a slope field, you calculate the slope for several points within a given window, like \([-2, 2] \times [-2, 2]\), and draw arrows accordingly. This gives you a snapshot of how the solution might look across the plane. Such visual representations are excellent aids in understanding the qualitative behavior of differential equations, even before obtaining an explicit solution.