Step by step
The question
Question: The circle has radius 10 m, measured with a possible error of 0.2 m.
Estimate the error in the area.
1 Draw and name
The radius is r = 10 m.
r = 10 m
The possible error in it is dr = 0.2 m: a small change.
dr = 0.2 m
2 Find dA
The area of a circle is A = π r².
A = πr2
Take the derivative, times dr: dA = 2πr dr.
dA = 2πr dr
The change in the area is about the derivative times the change in the radius.
dA = (derivative) × dr
3 Put the numbers in
Put in r = 10 and dr = 0.2.
dA = 2π(10)(0.2)
That is 4π, about 12.57 m².
= 4π ≈ 12.57 m²
Careful!
A small error in the radius makes a bigger error in the area.
Careful: dA is not dr: the error grows
4 Check
Let's check with the real change: the area at 10.2, minus the area at 10, is 12.69 m².
Close to our 12.57.
Right!
A(10.2) − A(10) = 12.69 m²
Let's look back.
First: draw and name.
Then: find dA.
Then: put the numbers in.
Last: check.
Answer
Careful: dA is not dr: the error grows