# Error analysis involving exponential

Given equation

$$E(t)=A \exp(-bt)$$

$A$ and $b$ are constant and $E$ is energy $t$ is time

If there is an error of say 1.5 percent In measured value of t

What is error in value of energy. How can I find it

• The error in $E$ is equal to $dE/dt$ multiplied by the error in $t$. – probably_someone Nov 15 '17 at 21:17

First and foremost, this looks like an error propagation problem. You are given an equation and some measurement of error.

Formal error propagation is given approximately by the formula (ignoring all covarriance):

$$\sigma_f^2 = \sum_i \sigma_{x_i}^2 \left(\frac{\partial f(x_i)}{\partial x_i} \right)^2$$

Where $\sigma_f$ is the total error that should be propagated, $\sigma_{x_i}$ is the error on the given varying element of $i$ (in your case, your $t$ variable), and $f(x_i)$ is the function of which you are trying to propagate error through (in your case, the equation $E(t) = A \exp\left( -bt \right)$.

Because you only have one term in your equation which has an error, namely $t$, such that $\sigma_t = 0.00015$, you can solve the error propagation formula for your new error.

The corresponding article of error propagation on Wikipedia has a much more detailed and formalized information about this subject. In particular is the section of pre-calculated error propagation formulas. Of interest is the following:

$$f = a \exp\left(bA\right) \qquad \Rightarrow \qquad \sigma_f^2 \approx f^2 \left(b \sigma_A \right)^2$$

Given the real value $A$, with error $\sigma_A$; the exactly known real-valued constants $a,b$ where $\sigma_a = \sigma_b = 0$. The ending value of the original function (see above) $f(x_i)$ given as $f$.

Please note that it is hard to do error propagation with the function that you provided without the value of the number itself, that is $t$ and its result, $f(t)$. For other functional arrangements, it may be easier.