Determine an expression for the uncertainty of the equivalent capacitance of this circuit.

Question

I know I use 1/C_eq = 1/C₁  + 1/C₂ + 1/C₃ 

 

ΔC_eq = ?

Answer 1

To find the uncertainty of the total you need to know the uncertainty of each element.  The uncertainty of the value of a circuit element is expressed by the elements tolerance - 10%, 5%, 2%, 1% etc.  You do not say what the tolerance is in the 3 capacitors.
 
Ahh - But, I see you want an expression and not a value.
But what kind of uncertainty are you looking for; a 1st, 2nd, 3rd approximation?
 
I tell how to obtain all of the different approximations to uncertainty at:
http://www.wyzant.com/resources/answers/39373/how_do_you_write_an_expression_of_uncertainty_of_the_spring_constant_by_propagation_of_errors_using_standard_deviation
 
You definitely do not have the statistical data to do a 4th approximation.
I suspect your professor is trying to get you to do a 3rd approximation of some type,
but you still do not have the data (tolerances) needed to do it properly but maybe we can get an expression.

There are divisions in the formula so you need to add and subtract percent values - right in line with tolerances which are given in percents.
But, there are also additions in the formula; and to do uncertainties with additions and subtractions you need to know the absolute values of the uncertainties. But we can calculate the absolute values if we know the values of the capacitors (C_x) and their tolerance (call it Tx).
 
So now let's look at your problem.
1/C_eq = 1/C₁  + 1/C₂ + 1/C₃ 

We can do it the way the equation sits but let's solve for C_eq - I think the solution might flow better.
C_eq = 1 / [ 1/C₁  + 1/C₂ + 1/C₃ ]

Working from the inside out as we do in math; we first need to handle the 1/C terms.
We are working with divisions so we need to add (or subtract) percent uncertainties - not an issue, we know them: T₁, T₂, T₃.
We need to add each to the numerators uncertainties. But what is the uncertainty of '1'? Zero of course. It is a mathematical constant. So the uncertainty of C₁ = T₁, and so on.

So now we have the uncertainty of the each item and we need to find the uncertainty of the sum [ 1/C₁  + 1/C₂ + 1/C₃ ] Here we need to add (or subtract) absolutes (let's call them A₁, A₂, A₃). We don't know the absolutes but we can calculate them A₁= T₁ x C₁.
Now almost the last step, we need to find the 1 / [ A₁ + A₂ + A₃ ]
Again the 1 is a constant with zero uncertainty.
But what is the percent uncertainty we need to add to it? What we have is the Absolute and strictly speaking we need to add the percent. But we are adding zero so does it really matter if we add it to the absolute or percent?

This leaves us with an absolute uncertainty of:
?C_eq-Abs = A₁ + A₂ + A₃ 

= (T₁ x C₁) + (T₂ x C₂) +(T₃ x C₃)

If you need the percent uncertainty to express it as a tolerance the just calculate the percent
%?C_eq = ?C_eq-Abs / C_eq