121616949-math.33 - y Δ x for our function y = f x = √...

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2 Instantaneous Rate Of Change: The Derivative Suppose that y is a function of x , say y = f ( x ). It is often necessary to know how sensitive the value of y is to small changes in x . EXAMPLE 2.1 Take, for example, y = f ( x ) = 625 - x 2 (the upper semicircle of radius 25 centered at the origin). When x = 7, we find that y = 625 - 49 = 24. Suppose we want to know how much y changes when x increases a little, say to 7.1 or 7.01. In the case of a straight line y = mx + b , the slope m = Δ y/ Δ x measures the change in y per unit change in x . This can be interpreted as a measure of “sensitivity”; for example, if y = 100 x + 5, a small change in x corresponds to a change one hundred times as large in y , so y is quite sensitive to changes in x . Let us look at the same ratio Δ
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Unformatted text preview: y/ Δ x for our function y = f ( x ) = √ 625-x 2 when x changes from 7 to 7 . 1. Here Δ x = 7 . 1-7 = 0 . 1 is the change in x , and Δ y = f ( x + Δ x )-f ( x ) = f (7 . 1)-f (7) = p 625-7 . 1 2-p 625-7 2 ≈ 23 . 9706-24 =-. 0294 . Thus, Δ y/ Δ x ≈ -. 0294 / . 1 =-. 294. This means that y changes by less than one third the change in x , so apparently y is not very sensitive to changes in x at x = 7. We say “apparently” here because we don’t really know what happens between 7 and 7 . 1. Perhaps y changes dramatically as x runs through the values from 7 to 7 . 1, but at 7 . 1 y 19...
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