Differentials continued
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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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Differentials continued

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 linedy = f’(x)dx

  • dy is often used to approximate y


Differentials continued

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


Example 1

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


Example 2

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


Example 2 continued

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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