Step by step
The question
Question: estimate √16.4 with a linear approximation.
√16.4 ≈ ?
The function is f(x) = √x.
Near 16.4 there is a friendly number: a = 16.
16 is a perfect square.
f(x) = √x
a = 16
1 The point
Put 16 into f.
We get 4.
The point is (16, 4).
f(16) = √16 = 4
2 The slope there
The slope is the derivative: f′(x) = 1/(2√x).
f′(x) = 12√x
At x = 16, the slope is 1/8.
f′(16) = 12√16 = 12 × 4 = 18
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) = 4 + 1/8 × (x − 16).
L(x) = 4 + 18(x − 16)
4 Estimate
Put 16.4 into L.
L(16.4) = 4 + 18(16.4 − 16)
Careful!
Use the distance from 16: 16.4 minus 16.
Not 16.4 itself.
Careful: not 4 + 18(16.4) ✗
16.4 minus 16 is 0.4.
= 4 + 18 × 0.4
1/8 times 0.4 is 0.05.
= 4 + 0.05
Add it up: 4.05.
= 4.05
So √16.4 ≈ 4.05.
√16.4 ≈ 4.05
5 Check
Let's check with a calculator: √16.4 is 4.04969 and a bit.
Our estimate 4.05 is very close.
Right!
calculator: √16.4 = 4.04969…
Let's look back.
First: the point.
Then: the slope there.
Then: the tangent line.
Then: estimate.
Last: check.
Answer
√16.4 ≈ 4.05