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Differentials ContinuedPowerPoint Presentation

Differentials Continued

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Differentials Continued

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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 linedy = f’(x)dx
- dy is often used to approximate y

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

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

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

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.