A(t)
is the amount of the investment at time t for the case of continuous compounding, write a differential equation satisfied by
A(t).
A(t) = Pe^nrt
A' = nrPe^nrt
dy/dx = nr y
To find the differential equation satisfied by A(t) for continuous compounding, we can use the formula for continuous compound interest.
The formula for continuous compound interest is given by:
A(t) = P * e^(rt)
Where:
A(t) is the amount of the investment at time t
P is the principal (initial investment)
e is the base of the natural logarithm (approximately equal to 2.71828)
r is the annual interest rate
t is the time in years
Now, let's differentiate both sides of the equation with respect to time (t):
(dA(t))/(dt) = (d/dt)(P * e^(rt))
Using the product rule of differentiation, we can differentiate the right side of the equation:
(dA(t))/(dt) = P * (d/dt)(e^(rt))
Now, let's find the derivative of e^(rt):
(d/dt)(e^(rt)) = re^(rt)
Substituting this derivative back into the equation, we have:
(dA(t))/(dt) = P * re^(rt)
Therefore, the differential equation satisfied by A(t) for continuous compounding is:
(dA(t))/(dt) = P * re^(rt)