Step by step
The question
Question: estimate √16.4 with a linear approximation.
There are four choices.
First we solve it, the normal way.
Then we pick.
The function is f(x) = √x.
Near 16.4 there is a friendly number: a = 16.
16 is a perfect square.
1 The point
Put 16 into f.
We get 4.
The point is (16, 4).
2 The slope there
The slope is the derivative: f′(x) = 1/(2√x).
Why: √x is x to the power one half.
The power rule: one half comes down, and the power drops by 1 to minus one half.
At x = 16, the slope is 1/8.
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) = 4 + 1/8 × (x − 16).
4 Estimate
Put 16.4 into L.
Careful!
Use the distance from 16: 16.4 minus 16.
Not 16.4 itself.
16.4 minus 16 is 0.4.
1/8 times 0.4 is 0.05.
Work it out: 4.05.
So √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!
6 Pick the choice
Our answer is √16.4 ≈ 4.05.
That is choice A.
7 Why not the others
B is 4.4.
Wrong: added 0.4: the slope at 16 is 1/8, not 1.
C is 4.2.
Wrong: used slope 1/2: it is 1/2√16.
D is 4.
Wrong: left out the change.
Let's look back.
First: the point.
Then: the slope there.
Then: the tangent line.
Then: estimate.
Then: check.
Then: pick the choice.
Last: why not the others.