Propagation of uncertainty when integrating or differentiating

Question

Lets say I have a polynomial $ax^4 +bx^3 +cx^2 +dx +e$ and the uncertainties on each coefficient. Now I need to calculate the tangent at some points as well as some areas under this curve. How would I find the uncertainties on those results?

My initial idea was to simply differentiate or integrate it and then use the partial derivatives method on the obtained expression(ignoring the error on "x" itself). ex. $slope = 4ax^3 + 3bx^2 +2cx +d$

$$\sigma_{slope}^2 = (4x^3)^2*\sigma_a^2 + (3x^2)^2*\sigma_b^2 + (2x)^2*\sigma_c^2 + \sigma_d^2$$

But higher x will make this error much higher which doesn't seem logical to me. I couldn't find the answer online and my book doesn't cover it.

Whats the right way to propagate uncertainties in such cases?

Answer 1

This is exactly the right approach if the errors in the coefficients are uncorrelated.

You would indeed expect the error of the slope, as an absolute value, to be larger when x is larger. An easy way to start to think about this is to consider a simple parabola: $ax^2$ with some error in a. In this case we find that

$\sigma_{slope} = 2x\sigma_a$

We know that at $x=0$ the slope is exactly zero, so the error should be zero. At some finite value of x we expect a finite slope and a finite error. In this simple case, the fractional error in the slope is constant, which to me makes intuitive sense:

$\frac{\sigma_{slope}}{slope} = \frac{\sigma_a}{a}$

Now if the errors in the coefficients are correlated, a more complete approach is necessary. As an example, if the polynomial coefficients were determined from fitting a set of measured data, the errors in the coefficients would be expected to be highly correlated.

In this case the formula for the propagation of error for a function $y = f(a,b)$ is:

$\sigma_y$ = $\left|\frac{\partial f}{\partial a}\right|^2\sigma_a^2 + \left|\frac{\partial f}{\partial b}\right|^2\sigma_b^2 + 2\frac{\partial f}{\partial a}\frac{\partial f}{\partial b}\small{\mathrm{COV}}_{ab}$

where $\small{\mathrm{COV}}_{ab}$ is the covariance between $a$ and $b$. This can be expanded to many parameters by including every cross term, which becomes quite long to write out. In your specific example you will have 10 cross terms to consider.