Solved Exercises In Slope Fields

CLAREMONT MCKENNA COLLEGE Solved Exercises on Slope Fields PART 1: TRADITIONAL METHOD 1) Sketch the slope field over the region [-5,5] x [-5,5]. Include the isocline in your sketch. y=x2+2, Next, trace the solution that satisfies �(1)=0 Solution: a) Understanding y (the Derivative): y represents the slope of the tangent line to the solution curve at any given point (x,y). For each point (x,y) in the plane, we compute y=f(x,y). b) Determine Slopes for Various Points: At each point (x,y), draw a small line segment with the slope given by y. If y=0 at a point, the line segment is horizontal. As y increases, the line segment becomes steeper. c) Plotting Slopes: Choose a grid of points in the (x,y) plane. Calculate the slope y at each grid point using the differential equation. For example, using a table of values for y, we substitute an arbitrary value for x (both negative and positive value) to determine the direction and behavior of the slope field. x -5 -4 -3 -2 -1 0 1 2 3 4 5 y 0 0 0 0 0 0 0 0 0 0 0 y 27 18 11 6 3 2 3 6 11 18 27d)