Step by step
The question
Question: estimate 3.1² with a linear approximation.
3.12 ≈ ?
The function is f(x) = x².
Near 3.1 there is a friendly number: a = 3.
f(x) = x2
a = 3
1 The point
Put 3 into f.
We get 9.
The point is (3, 9).
f(3) = 32 = 9
2 The slope there
The slope is the derivative: f′(x) = 2x.
f′(x) = 2x
At x = 3, the slope is 6.
f′(3) = 2(3) = 6
3 The tangent line
The tangent line: start at the height f of a, and go up by the slope times the distance from a.
L(x) = f(a) + f′(a)(x − a)
So L(x) = 9 + 6(x − 3).
L(x) = 9 + 6(x − 3)
4 Estimate
Put 3.1 into L.
L(3.1) = 9 + 6(3.1 − 3)
Careful!
Use the distance from 3: 3.1 minus 3.
Not 3.1 itself.
Careful: not 9 + 6(3.1) ✗
3.1 minus 3 is 0.1.
= 9 + 6 × 0.1
6 times 0.1 is 0.6.
= 9 + 0.6
Add it up: 9.6.
= 9.6
So 3.1² ≈ 9.6.
3.12 ≈ 9.6
5 Check
Let's check with a calculator: 3.1² is 9.61 and a bit.
Our estimate 9.6 is very close.
Right!
calculator: 3.12 = 9.61…
Let's look back.
First: the point.
Then: the slope there.
Then: the tangent line.
Then: estimate.
Last: check.
Answer
3.12 ≈ 9.6