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Unformatted text preview: MA 265 LECTURE NOTES: MONDAY, APRIL 21 Differential Equations Differences vs. Differentials. Consider a function x = x ( t ). Given two points P = ( t ,x ) and Q = ( t 1 ,x 1 ) on the graph of this function, we can draw a secant line through them. The slope of this line is Slope of Secant Line at P = Δ x Δ t ( P ) = x 1 x t 1 t = x ( t + Δ t ) x ( t ) Δ t where Δ t = t 1 t is a difference . As P and Q move closer to each other, we can draw a line tangent to the curve x = x ( t ). The slope of this line is Slope of Tangent Line at P = dx dt ( P ) = lim Q → P x 1 x t 1 t = lim Δ t → x ( t + Δ t ) x ( t ) Δ t where dt is a differential . It is best to think of a differential as an infinitesimally small difference – but this quantity is nonzero! Any equation that involves differences Δ x and Δ t is called a difference equation , and any equation that involves differentials dx and dt is called a differential equation . Example. Let x = x ( t ) denote the size of a population at time t . How quickly a population grows depends on the number of people present in the population; these people may be engineers, doctors, etc. to help construct society. We see that the rate of change of growth is proportional to the number of people present:construct society....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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