A ban pays interest by continuous compounding, that is, by treating the interest rate as the instantaneous rate of change of principal. Suppose in an account interest accrues at 8% per year, compounded continuously. Calculate the percentage increase in such an account over one year.
[Take e0.08≈1.0833]
Let the principal, rate and time be Rs P, r and t years.
Also, let the initial principal be Po.
⇒ 
Integrating both sides, we have
⇒ ∫
∫dt
⇒ log|P| = 
t + c……(1)
Now, at t = 0, P = Po
log| Po | = 0 + c
⇒ c = log| Po |……(2)
Putting the value of c in equation (1) we have,
log|P| = 
t + log|Po|
⇒ log|P| – log|Po| = 
t
⇒ (log |P| – log|Po|) = 
t [
]
⇒ log (
= 
t ……(3)
Now, t = 1 year, r = 8%
∴ log (
= 
×1
⇒ log (
= 0.08
⇒ 
⇒ 
⇒ 
(Given: 
= 1.0833)
⇒ 
∴ Percentage increase = 0.0833×100 = 8.33%