Final answer:
To plot the slope field for y′ = 1/2x + 1, draw short line segments with the given slopes at various points. Then, sketch integral curves starting from initial conditions y(0) = -2, -1, 0, 1, 2, following the direction of the slopes in the field.
Step-by-step explanation:
The question involves plotting a slope field and sketching integral curves for the differential equation y′ = 1/2x + 1, for given initial conditions y(0) = −2, −1, 0, 1, 2. A slope field is a graphical representation of the slopes of a differential equation at a given set of points. To sketch a slope field, you draw short line segments with the slope given by the differential equation at various points in the xy-plane.
First, you would plot the points where the slope field will be drawn. The slope of these line segments at any point (x, y) will be 1/2x + 1. For example, at x = 0, the slope m = 1. At x = 2, the slope m = 2, and so on for the slopes −2, −1, 0, 1, and 2. Next, you draw line segments representing these slopes at multiple points along the x-axis.
Once the slope field is drawn, you then sketch integral curves, which are potential solutions to the differential equation that also satisfy the initial conditions given. For an initial condition like y(0) = 0, you would start your curve at the point (0, 0) and follow the direction of the slopes in the field. Repeat this for the initial conditions y(0) = −2, −1, 0, 1, 2, tracing out curves that start at (0, −2), (0, −1), (0, 0), (0, 1), and (0, 2), respectively.