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Differentials Continued

Differentials Continued. Comparing dy and  y  y is the actual change in y from one point to another on a function (y 2 – y 1 ) d y is the corresponding change in y on the tangent line dy = f’(x) dx d y is often used to approximate  y. Given points P and Q on function f.

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Differentials Continued

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  1. Differentials Continued Comparing dyand y • y is the actual change in y from one point to another on a function (y2 – y1) • dy is the corresponding change in y on the tangent line dy = f’(x)dx • dy is often used to approximate y

  2. Given points P and Q on function f. The slope between P and Q is y / x The slope of the tangent line at P is dy/dx Notice: dx = xand dy ≈ y Q P

  3. Example 1 Find both y and dy and compare. y = 1 – 2x2at x = 1 when x= dx = -.1 Solution: y = f(x + x) – f(x) = f(.9) – f(1) = -.62 – (-1) = .38 dy = f’(x)dx = (-4)(-.1) = .4 ** Note that y ≈ dy

  4. Example 2 The radius of a ball bearing is measured to be .7 inches. If the measurement is correct to within .01 inch, estimate the propagated error in the volume V of the ball bearing. ** Propagated error means the resulting change, or error, in measurement

  5. Example 2 Continued To decide whether the propagated error is small or large, it is best looked at relative to the measurement being calculated. • Find the relative error in volume of the ball bearing. ** Relative error is dy/y, or in this case dV/V • Find the percent error,(dV/V)*100.

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