Step by step
The question
Question: estimate √8.9 with a linear approximation.
√8.9 ≈ ?
The function is f(x) = √x.
Near 8.9 there is a friendly number: a = 9.
9 is a perfect square.
f(x) = √x
a = 9
1 The point
Put 9 into f.
We get 3.
The point is (9, 3).
f(9) = √9 = 3
2 The slope there
The slope is the derivative: f′(x) = 1/(2√x).
f′(x) = 12√x
At x = 9, the slope is 1/6.
f′(9) = 12√9 = 12 × 3 = 16
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) = 3 + 1/6 × (x − 9).
L(x) = 3 + 16(x − 9)
4 Estimate
Put 8.9 into L.
L(8.9) = 3 + 16(8.9 − 9)
Careful!
Use the distance from 9: 8.9 minus 9.
Not 8.9 itself.
Careful: not 3 + 16(8.9) ✗
8.9 minus 9 is −0.1.
= 3 − 16 × 0.1
1/6 times 0.1 is about 0.01666.
≈ 3 − 0.01666
Add it up: 2.98333.
≈ 2.98333
So √8.9 ≈ 2.98333.
√8.9 ≈ 2.98333
5 Check
Let's check with a calculator: √8.9 is 2.98329 and a bit.
Our estimate 2.98333 is very close.
Right!
calculator: √8.9 = 2.98329…
Let's look back.
First: the point.
Then: the slope there.
Then: the tangent line.
Then: estimate.
Last: check.
Answer
√8.9 ≈ 2.98333