1.2: DIRECTION FIELDS In Math 3B, we think a lot about finding an unknown function F(t) Summary: §1.2: DIRECTION FIELDS In Math 3B, we think a lot about finding an unknown function F(t) whose derivative is known, some given function f(t). One approach is geometric. At each point in the plane, we know the slope of the function F(t), so we can plot a little segment with that slope, and cover the whole plane. This gives a picture of what the graph of f(t) should look like. The top of Figure 1 shows this for f(t) = t. Convince yourself of this: the slopes are large when the t coordinate is large and small when the t coordinate is small. The slopes do not depend on the y coordinate at all. We can see solutions by following the slopes. They are parabolas just as we expect, since F(t) = t2 /2 + a constant. This idea occurs in Math 3B but is not emphasized. In this course, the derivative of the unknown function depends on both the t and y coordinates. The same idea as above gives the DIRECTION FIELD or SLOPE FIELD for the differential equation. In general it does not have the symmetry under vertical translation that the 3B examples had, so the problems are harder now. The bottom of Figure 1 shows the direction field for Collections: Mathematics