Step by step
The question
Question: estimate √9.2 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 9.2 there is a friendly number: a = 9.
9 is a perfect square.
1 The point
Put 9 into f.
We get 3.
The point is (9, 3).
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 = 9, the slope is 1/6.
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.
So L(x) = 3 + 1/6 × (x − 9).
4 Estimate
Put 9.2 into L.
Careful!
Use the distance from 9: 9.2 minus 9.
Not 9.2 itself.
9.2 minus 9 is 0.2.
1/6 times 0.2 is about 0.03333.
Work it out: 3.03333.
So √9.2 ≈ 3.03333.
5 Check
Let's check with a calculator: √9.2 is 3.03315 and a bit.
Our estimate 3.03333 is very close.
Right!
6 Pick the choice
Our answer is √9.2 ≈ 3.03333.
That is choice D.
7 Why not the others
A is 3.2.
Wrong: added 0.2: the slope at 9 is 1/6, not 1.
B is 3.1.
Wrong: used slope 1/2: it is 1/2√9.
C is 3.
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.