May 28th, 2015
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DIFFERENTIALSLINEAR APPROXIMATIONFirst: Consider howPoint-Slope: is the same asTangent Line: So we will now use this idea to build an equation of a LINE which will approximate a FUNCTION.EXAMPLE 1:Find a tangent line approximation at ( 0 , 1 ) then see how good it is So, the linear approximation equation is: or So, is the same as but only for values very close to x = 0. We would have to create a completely different line for a different (x, y) pair.x-0.5-0.1-0.0100.010.1.5.521.9002.990000211.00999981.09981.479.5.9.9911.011.11.5Error-.021-.0002-.00000020.0000002.0002.021EXAMPLE 1A: You Try It: Create a linear approximation equation for the function: at GENERAL COMPARISON BETWEEN y AND dy AQ = the real change in y = y = AB = the fake approximation of y = dy known as the differential of y derived from the slope of the tangent line mtan = the derivative = isolate the dy now So we have it this is the definition of a DIFFERENTIAL= the differential of yEXAMPLE 2: Comparing y and dyLet . Find dy when x = 1 and dx = .01. Compare this to y for the same x, and x.To find dy, we simply need to use the differential form of the derivative expression A.K.A. the definition of a differential.To compare this to the actual y, the real change in y, like done in an algebra course we nee

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