Step by step
The question
Question: The square has side 8 m, measured with a possible error of 0.1 m.
Estimate the error in the area.
1 Draw and name
The side is s = 8 m.
s = 8 m
The possible error in it is ds = 0.1 m: a small change.
ds = 0.1 m
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 = 8 and ds = 0.1.
dA = 2(8)(0.1)
So the error in the area is about 1.6 m².
= 1.6 m²
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 8.1, minus the area at 8, is 1.61 m².
Close to our 1.6.
Right!
A(8.1) − A(8) = 1.61 m²
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