multivariable_02_Instantaneous_Rate_of_Change-_The_Derivative_4up

Multivariable_02_Instantaneous_Rate_of_Change-_The_Derivative_4up

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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 ) = radicalbig 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 y/ Δ x for our function y = f ( x ) = radicalbig 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) = radicalbig 625 - 7 . 1 2 - radicalbig 625 - 7 2 23 . 9706 - 24 = - 0 . 0294 . Thus, Δ y/ Δ x ≈ - 0 . 0294 / 0 . 1 = - 0 . 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 just happens to be close to its value at 7. This is not in fact the case for this particular function, but we don’t yet know why. 17 18 Chapter 2 Instantaneous Rate of Change: The Derivative One way to interpret the above calculation is by reference to a line. We have computed the slope of the line through (7 , 24) and (7 . 1 , 23 . 9706), called a chord of the circle. In general, if we draw the chord from the point (7 , 24) to a nearby point on the semicircle (7 + Δ x, f (7 + Δ x )), the slope of this chord is the so-called difference quotient slope of chord = f (7 + Δ x ) - f (7) Δ x = radicalbig 625 - (7 + Δ x ) 2 - 24 Δ x . For example, if x changes only from 7 to 7.01, then the difference quotient (slope of the chord) is approximately equal to (23 . 997081 - 24) / 0 . 01 = - 0 . 2919. This is slightly less steep than the chord from (7 , 24) to (7 . 1 , 23 . 9706). As the second value 7 + Δ x moves in towards 7, the chord joining (7 , f (7)) to (7 + Δ x, f (7 + Δ x )) shifts slightly. As indicated in figure 2.1, as Δ x gets smaller and smaller, the chord joining (7 , 24) to (7 + Δ x, f (7 + Δ x )) gets closer and closer to the tangent line to the circle at the point (7 , 24). (Recall that the tangent line is the line that just grazes the circle at that point, i.e., it doesn’t meet the circle at any second point.) Thus, as Δ x gets smaller and smaller, the slope Δ y/ Δ x of the chord gets closer and closer to the slope of the tangent line. This is actually quite difficult to see when Δ x is small, because of the scale of the graph. The values of Δ x used for the figure are 1, 5, 10 and 15, not really very small values. The tangent line is the one that is uppermost at the right hand endpoint.
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