Repeating this process 8 more times gives us our slope field.

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Presentation transcript:

Repeating this process 8 more times gives us our slope field. dy/dx = y - 1 x2 Part (a) (1/4) (-1/4) Let’s start with a random point—(2,2) for example... dy/dx = 2 - 1 22 Repeating this process 8 more times gives us our slope field. m = 1/4

Separate the variables dy/dx = y - 1 x2 Part (b) (y-1)-1 = x-2 dx Separate the variables ln y-1 = -x-1 + C Integrate both sides ln 0-1 = -2-1 + C Find the value of C 0 = -1/2 + C C = 1/2 ln y-1 = -x-1 + 1/2 ( + ½) -1 x ln y-1 e = Exponentiate both sides

Part (b) ( + ½) e =  1 ( + ½) e = y-1 ( + ½) e = y-1  e0 =  1 1 ( + ½) -1 2 e =  1 ( + ½) -1 x e = y-1 ( + ½) -1 x e = y-1  e0 =  1 1 Use the negative part. ( + ½) -1 x e = y  1 ( + ½) -1 x e = y 1 - ( + ½) -1 x ln y-1 e = e

Part (c) ( + ½) e 1 - lim = 1 – e0+½ = 1 – e½ 1 – e = ( + ½) e = y 1 - ( + ½) -1 x e 1 - lim x  = 1 – e0+½ = 1 – e½ 1 – e = ( + ½) -1 x e = y 1 -