121616949-math.36 - Starting with one function √ 625-x 2...

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22 Chapter 2 Instantaneous Rate Of Change: The Derivative example, the “24” in the calculation came from 625 - 7 2 , so we’ll need to fix that too. p 625 - ( x + Δ x ) 2 - 625 - x 2 Δ x = = p 625 - ( x + Δ x ) 2 - 625 - x 2 Δ x p 625 - ( x + Δ x ) 2 + 625 - x 2 p 625 - ( x + Δ x ) 2 + 625 - x 2 = 625 - ( x + Δ x ) 2 - 625 + x 2 Δ x ( p 625 - ( x + Δ x ) 2 + 625 - x 2 ) = 625 - x 2 - 2 x Δ x - Δ x 2 - 625 + x 2 Δ x ( p 625 - ( x + Δ x ) 2 + 625 - x 2 ) = Δ x ( - 2 x - Δ x ) Δ x ( p 625 - ( x + Δ x ) 2 + 625 - x 2 ) = - 2 x - Δ x p 625 - ( x + Δ x ) 2 + 625 - x 2 Now what happens when Δ x is very close to zero? Again it seems apparent that the quotient will be very close to - 2 x 625 - x 2 + 625 - x 2 = - 2 x 2 625 - x 2 = - x 625 - x 2 . Replacing x by 7 gives - 7 / 24, as before, and now we can easily do the computation for 12 or any other value of x between - 25 and 25. So now we have a single, simple formula, - x/ 625 - x 2 , that tells us the slope of the tangent line for any value of x . This slope, in turn, tells us how sensitive the value of y is to changes in the value of x . What do we call such a formula? That is, a formula with one variable, so that substi-
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Unformatted text preview: Starting with one function, √ 625-x 2 , we have derived, by means of some slightly nasty algebra, a new function,-x/ √ 625-x 2 , that gives us important information about the original function. This new function in fact is called the derivative of the original func-tion. If the original is referred to as f or y then the derivative is often written f ± or y ± and pronounced “f prime” or “y prime”, so in this case we might write f ± ( x ) =-x/ √ 625-x 2 . At a particular point, say x = 7, we say that f ± (7) =-7 / 24 or “ f prime of 7 is-7 / 24” or “the derivative of f at 7 is-7 / 24.” To summarize, we compute the derivative of f ( x ) by forming the difference quotient f ( x + Δ x )-f ( x ) Δ x ,...
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