Step by step
The question
Question: estimate 1.02⁴ with a linear approximation.
The function is f(x) = x⁴.
Near 1.02 there is a friendly number: a = 1.
1 The point
Put 1 into f.
We get 1.
The point is (1, 1).
2 The slope there
The slope is the derivative: f′(x) = 4x³.
At x = 1, the slope is 4.
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) = 1 + 4(x − 1).
Let's see it.
The curve.
And the tangent line at x = 1.
Near 1, they are almost the same.
4 Estimate
Put 1.02 into L.
Careful!
Use the distance from 1: 1.02 minus 1.
Not 1.02 itself.
1.02 minus 1 is 0.02.
4 times 0.02 is 0.08.
Add it up: 1.08.
So 1.02⁴ ≈ 1.08.
5 Check
Let's check with a calculator: 1.02⁴ is 1.08243 and a bit.
Our estimate 1.08 is very close.
Right!
Let's look back.
First: the point.
Then: the slope there.
Then: the tangent line.
Then: estimate.
Last: check.