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1.7 Coaxing the Derivative out of NOTHING

# 1.7 Coaxing the Derivative out of NOTHING - [1.7 Coaxing...

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[1.7] Coaxing the Derivative out of NOTHING! First we start with ( ) y f x and what we know about AROC at x a slope of the secant from ( , ( )) to ( , ( ))= ( ) ( ) AROC of x a a f a a h f a h f a h f a y f h x Now As h gets smaller and smaller and smaller the length of the secant line gets ever smaller, but at the same time the slope of the secant line gets ever closer to the slope of the tangent line. E ven though we don’t really know the value of the slope of the tange nt line we know the secant line slope is getting closer to it. For small the slope of the secant from ( ) ( ) Slope of the tangent to at the input x h f a h f a f h Now for zee Magic…Vee do zee CALCULUS und The secant line disappears, thus ceases to exist and therefore has no slope at the same “moment” that it s slope becomes the slope of the tangent line. We say the AROC is no longer
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