MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 14 Basic Mathematical Models (cont’d) Slope Fields. It is not necessary to fnd the exact solution oF a di±erential equation in order to understand how the solution behaves. We discuss this concept through several examples. Consider a plot oF x ( t )versus t i.e. we plot the position x on the vertical axis and time t on the horizontal axis. Through any two points on the curve we can draw a line segment. Recall that given two distinct points ( t0 ,x0 ) and ( t 1 ,x 1 ), the line through them has slope slope oF secant line = ∆ x ∆ t = x ( t 1 ) − x ( t0 ) t 1 − t0 . OF course, iF the points are close to each other, we fnd the slope oF the tangent line: slope oF tangent line = lim t 1 → t0 x ( t 1 ) − x ( t0 ) t 1 − t0 = dx dt ( t0 ) . In particular, the slope oF the secant line is an approximation oF the slope oF the tangent line: ∆ x ∆ t ≈ dx dt = ⇒ ∆ x ≈ dx dt ∆ t. Conversely, iF we knew the slope x ° ( t ) at every point ( t, x ), we could do a reasonable job oF sketching the path x ( t ) vs. t. This is called the
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