The sales, in millions of dollars, of a laser disc recording of a hit movie t years from the date of release is given by:
S(t) = (5t)/(t^2 + 1)
a) Find the rate at which the sales are changing at time t.
b) How fast are the sales changing at the time the laser discs are released?
a)
S'(t) = dS(t)/dt
S'(t) = ((d5t/dt)(t2+1) - 5t d(t2+1)/dt)/(t2+1)2
S'(t) = (5(t2+1) - 10t2)/(t2+1)2
S'(t) = (5-5t2)/(t4+2t2+1)
To find the rate at which the sales are changing at time t, we need to find the derivative of the function S(t) with respect to t.
a) Calculating the derivative of S(t):
Step 1: Start with the function S(t).
S(t) = (5t)/(t^2 + 1)
Step 2: Use the quotient rule to find the derivative.
The quotient rule is applied as follows:
If we have a function f(x) = u(x)/v(x), where both u(x) and v(x) are functions of x, then the derivative of f(x) is given by:
f'(x) = (v(x) * u'(x) - u(x) * v'(x)) / (v(x))^2
In our case:
u(t) = 5t
v(t) = t^2 + 1
Now we can calculate the derivative:
S'(t) = [(t^2 + 1) * (5) - (5t) * (2t)] / (t^2 + 1)^2
Simplifying further, we get:
S'(t) = (5t^2 + 5 - 10t^2) / (t^2 + 1)^2
= (-5t^2 + 5) / (t^2 + 1)^2
Therefore, the rate at which the sales are changing at time t is given by:
S'(t) = (-5t^2 + 5) / (t^2 + 1)^2
b) To find how fast the sales are changing at the time the laser discs are released, we can substitute t=0 into the derivative equation we obtained in part a):
S'(0) = (-5(0)^2 + 5) / ((0)^2 + 1)^2
= 5 / 1
= 5
Therefore, the sales are changing at a rate of 5 million dollars per unit time when the laser discs are released.