Estimate e^0.1 with a linear approximation Step by Step

Step by step

The question

Question: estimate e^(0.1) with a linear approximation.

e^(0.1) ≈ ?

The function is f(x) = eˣ.

Near 0.1 there is a friendly number: a = 0.

e to the 0 is 1.

f(x) = e^x

a = 0

1 The point

Put 0 into f.

We get 1.

The point is (0, 1).

f(0) = e⁰ = 1

2 The slope there

The slope is the derivative: f′(x) = eˣ.

f′(x) = e^x

At x = 0, the slope is 1.

f′(0) = e⁰ = 1

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 + x.

L(x) = 1 + x

Let's see it.

The curve.

And the tangent line at x = 0.

Near 0, they are almost the same.

4 Estimate

Put 0.1 into L.

L(0.1) = 1 + (0.1 − 0)

Careful!

Use the distance from 0: 0.1 minus 0.

Not 0.1 itself.

Careful: not 1 + (0.1) ✗

0.1 minus 0 is 0.1.

= 1 + 0.1

Add it up: 1.1.

= 1.1

So e^(0.1) ≈ 1.1.

e^(0.1) ≈ 1.1

5 Check

Let's check with a calculator: e^(0.1) is 1.10517 and a bit.

Our estimate 1.1 is very close.

Right!

calculator: e^(0.1) = 1.10517…

Let's look back.

First: the point.

Then: the slope there.

Then: the tangent line.

Then: estimate.

Last: check.

Answer