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.
Comparing dyand y
The slope between P and Q is
y / x
The slope of the tangent line at P is
dx = xand
dy ≈ y
Find both y and dy and compare.
y = 1 – 2x2at x = 1 when x= dx = -.1
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.
** Relative error is dy/y, or in this case dV/V