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lecture_2 (dragged) - MA 36600 LECTURE NOTES: WEDNESDAY,...

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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 ( t 0 ,x 0 ) and ( t 1 ,x 1 ), the line through them has slope slope oF secant line = x t = x ( t 1 ) x ( t 0 ) t 1 t 0 . 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 t 0 x ( t 1 ) x ( t 0 ) t 1 t 0 = dx dt ( t 0 ) . 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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