Step by step
The question
Question: The square has side 20 cm, measured with a possible error of 0.5 cm.
Estimate the error in the area.
1 Draw and name
The side is s = 20 cm.
s = 20 cm
The possible error in it is ds = 0.5 cm: a small change.
ds = 0.5 cm
2 Find dA
The area of a square is A = s².
A = s2
Take the derivative, times ds: dA = 2s ds.
dA = 2s ds
The change in the area is about the derivative times the change in the side.
dA = (derivative) × ds
3 Put the numbers in
Put in s = 20 and ds = 0.5.
dA = 2(20)(0.5)
So the error in the area is about 20 cm².
= 20 cm²
Careful!
A small error in the side makes a bigger error in the area.
Careful: dA is not ds: the error grows
4 Check
Let's check with the real change: the area at 20.5, minus the area at 20, is 20.25 cm².
Close to our 20.
Right!
A(20.5) − A(20) = 20.25 cm²
Let's look back.
First: draw and name.
Then: find dA.
Then: put the numbers in.
Last: check.
Answer
Careful: dA is not ds: the error grows