Question:

Consider the differential equation dy/dx = x + 2.

1. Compute the value of the derivative at x = 1.
2. Sketch the slope field for the given differential equation on the interval -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2.
3. Using the slope field, estimate the solution to the differential equation that satisfies the initial condition y(0) = -1.
4. Find the exact solution to the differential equation with the given initial condition.

Answer:

1. To compute the value of the derivative at x = 1, substitute x = 1 into the differential equation dy/dx = x + 2:

   dy/dx = x + 2 dy/dx = 1 + 2 dy/dx = 3

   Therefore, the value of the derivative at x = 1 is 3.

2. To sketch the slope field, we plot small line segments at various points on the x-y plane based on the slope at those points. The slope at each point (x, y) is given by x + 2.

   The following slope field is obtained:

   

3. To estimate the solution to the differential equation that satisfies the initial condition y(0) = -1, we use the slope field. Starting from the point (0, -1), we follow the slope field lines to sketch the approximate solution curve.

   Starting from (0, -1), at x = 0, the slope is 2. We move upward by 2 units to reach the next point. At x = 1, the slope is 3. We move upward by 3 units to reach the next point. Continuing this process, we sketch the approximate solution curve as follows:

   

   Based on the slope field, the approximate solution to the differential equation that satisfies the initial condition y(0) = -1 is shown by the curve above.

4. To find the exact solution to the differential equation with the given initial condition, we need to integrate both sides of the equation.

   dy/dx = x + 2

   ∫dy = ∫(x + 2)dx

   Integrating both sides, we get:

   y = (1/2)x^2 + 2x + C

   Now, using the initial condition y(0) = -1, we can solve for the constant C:

   -1 = (1/2)(0)^2 + 2(0) + C -1 = C

   Therefore, the exact solution to the differential equation with the initial condition y(0) = -1 is:

   y = (1/2)x^2 + 2x - 1