Estimating with Differentials

EXAMPLE 5 Finding Differentials of Functions
(a)
(b)
Estimating with Differentials
Suppose we know the value of a differentiable function ƒ(x) at a point a and want to predict
how much this value will change if we move to a nearby point If dx is small,
then we can see from Figure 3.51 that is approximately equal to the differential dy.
Since
the differential approximation gives
where Thus the approximation can be used to calculate
when ƒ(a) is known and dx is small.
EXAMPLE 6 Estimating with Differentials
The radius r of a circle increases from to 10.1 m (Figure 3.52). Use dA to estimate
the increase in the circle’s area A. Estimate the area of the enlarged circle and compare
your estimate to the true area.
Solution Since the estimated increase is
Thus,
The area of a circle of radius 10.1 m is approximately
The true area is
The error in our estimate is which is the difference
Error in Differential Approximation
Let ƒ(x) be differentiable at and suppose that is
Differential Estimates of Change
In Exercises 37–42, write a differential formula that estimates the
given change in volume or surface area.
37. The change in the volume of a sphere when the radius
changes from to
38. The change in the volume of a cube when the edge
lengths change from to
39. The change in the surface area of a cube when the edge
lengths change from to
40. The change in the lateral surface area of a
right circular cone when the radius changes from to
and the height does not change
41. The change in the volume of a right circular cylinder
when the radius changes from to and the height does
not change
42. The change in the lateral surface area of a right circular
cylinder when the height changes from to and the radius
does not change