Step by step
The question
Question: estimate √4.1 with a linear approximation.
The function is f(x) = √x.
Near 4.1 there is a friendly number: a = 4.
4 is a perfect square.
1 The point
Put 4 into f.
We get 2.
The point is (4, 2).
2 The slope there
The slope is the derivative: f′(x) = 1/(2√x).
At x = 4, the slope is 1/4.
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.
So L(x) = 2 + 1/4 × (x − 4).
Let's see it.
The curve.
And the tangent line at x = 4.
Near 4, they are almost the same.
4 Estimate
Put 4.1 into L.
Careful!
Use the distance from 4: 4.1 minus 4.
Not 4.1 itself.
4.1 minus 4 is 0.1.
1/4 times 0.1 is 0.025.
Add it up: 2.025.
So √4.1 ≈ 2.025.
5 Check
Let's check with a calculator: √4.1 is 2.02485 and a bit.
Our estimate 2.025 is very close.
Right!
Let's look back.
First: the point.
Then: the slope there.
Then: the tangent line.
Then: estimate.
Last: check.