Question

The total charge in an electrical circuit as a function of time is given by q=t/t^2+1 coulombs. Find the maximum charge q.
Please provide your reasoning.

Answer 1

q = t/(t^2 +1) = t•(t^2 +1)^-1.
The derivative of this expression is
dq/dt = 1•(t^2+1)^-1 - t•(t^2 +1)^-2•2•t = 1/(t^2+1) -2•t^2/(t^2 + !)^2 =
=(t+1-2•t^2)/(t^2+1)^2 .
The maximum of this function at
(t+1-2•t^2)/(t^2+1)^2 = 0,
t+1-2•t^2= 0
2•t^2 – t -1 = 0.
Two roots: t1 =-0.5 (impossible for time), t2 = 1 s.
q = t/(t^2 +1) = 1/(1+1) = 0.5 Coulomb.

Answer 2

Thank you!!