Calculus I:
functions of several variables
linear approximation
Error calculation for
single variable function

Error calculation for
a function of single variable

When we do measurements we make errors that are due to the precision of the tools we have.

We must therefore make a distinction between exact value (theoretical) and approximate value (obtained by practice).

For example, suppose we measure the dimension of a square.

We obtain by a measure 4.97 meters, and we know that our measuring devices give a precision of 0.05 meters.

However, the exact measurement is 5 meters.

There is therefore a difference between measured value and exact value. In our case, this error is 0.03 meters.


Definition 1

In general, x is the measured value, δx is the error of the made measurement and Δx the precision of the measuring device. Thus, the exact value is x + δx. In addition, w have |δx|≤ Δx.

In our case, we have x = 4.97 and δx = + 0.03 and Δx = 0.05.

In practice, only x and Δx are known.

We notice that in practice we do not know δx .

Of course, if we know a measurement and the error of this measure then we will know the exact value.

Now we want to evaluate the surface of the square. As in practice , the exact value x + δx, is unknown, the only thing we can do is to use x, the measured value.

As our measure gave us a length of 4.97 meters, we calculate f(4.97), where f (t) = t².
We have f(4.97) = (4.97)² = 24.7009.

This value does not match the exact area of the square, because we have used for the calculations an approximate value.

We cannot do better more than that. Indeed, in practice we do not know the exact measure of the dimension !

The only thing we can do is estimate the error made by taking an approximate value. This error is f(x) - f(x + δ_x).

We know the following approximation:

f (x + δx) ≈ f(x) + f'(x)δ_x , that gives :

f (x + δx) - f(x) ≈ f'(x)δ_x

We only know an increase of |δ_x| which is Δx = 0.05. This gives :

|f(x + δx) - f(x)| ≈ |f'(x)|Δx

In our case x = 4.97 and Δx = 0. 05. As f'(x) = 2.x, we get f'(4.97) = 2 x 4.97 = 9.94, it comes then:

|f(x + δx) - f(x)| ≈ 9.94 x 0.05 = 0.497.

The error is therefore about 0.497 m2.

This means that in our case, the figures given after the decimal point in the calculation of f(4.97) have no significance for the estimation of the measured area.

In order to ease the writing, we will introduce a notation.


Definition 2

The error |f(x + δx) - f(x)| is noted δf. This error is also called absolute error.

Proposition

We denote |δf| by Δf, the order of magnitude of the absolute error. So

Δf ≈ |f'(x)| Δx , called : the absolute error .

In the foregoing we have studied the error Δf . We have seen that this error was about 0.5 m2. Has a good approximation been obtained?

Indeed, there is a significant difference between an error of 0.5 m2 on a surface of 1.000 m2 and on a surface of 2 m2 .

To find out if an error is big or not, we look at what proportion, what percentage it represents with respect to f (x).

Definition 3

The Relative error is the quotient: Δf/|f(x)| . This number is expressed in %.

In the previous example, the relative error is: Δf/|f(x)| = 0.497 /|f(4.97)| = 0.497 / 24.7009 = 0.02006 = 2%

Note : Calculate the relative error is to calculate

(|f'(x)|/|f(x)|)Δx

So we can calculate the relative error from a logarithmic derivative calculation.