Use a linear approximation to estimate 1/2.1 Step by Step

Step by step

The question

Question: estimate 1/2.1 with a linear approximation.

12.1 ≈ ?

The function is f(x) = 1 over x.

Near 2.1 there is a friendly number: a = 2.

f(x) = 1x

a = 2

1 The point

Put 2 into f.

We get 1/2.

The point is (2, 1/2).

f(2) = 12

2 The slope there

The slope is the derivative: f′(x) = −1/x².

f′(x) = −1x²

At x = 2, the slope is −1/4.

f′(2) = −12² = −14

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) = 1/2 − 1/4 × (x − 2).

L(x) = 12 − 14(x − 2)

Let's see it.

The curve.

And the tangent line at x = 2.

Near 2, they are almost the same.

4 Estimate

Put 2.1 into L.

L(2.1) = 12 − 14(2.1 − 2)

Careful!

Use the distance from 2: 2.1 minus 2.

Not 2.1 itself.

Careful: not 12 − 14(2.1) ✗

2.1 minus 2 is 0.1.

= 12 − 14 × 0.1

1/4 times 0.1 is 0.025.

= 12 − 0.025

Add it up: 0.475.

= 0.475

So 1/2.1 ≈ 0.475.

12.1 ≈ 0.475

5 Check

Let's check with a calculator: 1/2.1 is 0.47619 and a bit.

Our estimate 0.475 is very close.

Right!

calculator: 12.1 = 0.47619…

Let's look back.

First: the point.

Then: the slope there.

Then: the tangent line.

Then: estimate.

Last: check.

Answer