Step by step
The question
Question: The circle has radius 8 cm, measured with a possible error of 0.1 cm.
Estimate the error in the area.
1 Draw and name
The radius is r = 8 cm.
r = 8 cm
The possible error in it is dr = 0.1 cm: a small change.
dr = 0.1 cm
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 = 8 and dr = 0.1.
dA = 2π(8)(0.1)
That is 1.6π, about 5.03 cm².
= 1.6π ≈ 5.03 cm²
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 8.1, minus the area at 8, is 5.06 cm².
Close to our 5.03.
Right!
A(8.1) − A(8) = 5.06 cm²
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