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)=[{4t+1}/{t^3+1}]
How fast is the sales changing one year the laser discs are released?
substitute t=1 in
S(t)=[{4t+1}/{t^3+1}]
to get
S(1)=[{4*1+1}/{1^3+1}]
Can you evaluate S(1)?
To find how fast the sales are changing one year after the laser discs are released, we need to find the derivative of the sales function S(t) with respect to time t.
The derivative of S(t) can be found using the quotient rule. The quotient rule states that if we have a function of the form f(t) = g(t) / h(t), where g(t) and h(t) are differentiable functions, then the derivative of f(t) is given by:
f'(t) = [g'(t) * h(t) - g(t) * h'(t)] / [h(t)]^2
In this case, g(t) = 4t + 1 and h(t) = t^3 + 1. Let's find the derivatives of g(t) and h(t).
g'(t) = 4 (since the derivative of 4t is 4, and the derivative of a constant term 1 is 0)
h'(t) = 3t^2 (since the derivative of t^3 is 3t^2, and the derivative of a constant term 1 is 0)
Now we can substitute these derivatives into the quotient rule formula:
S'(t) = [(4)(t^3 + 1) - (4t + 1)(3t^2)] / [(t^3 + 1)]^2
Simplifying, we get:
S'(t) = (4t^3 + 4 - 12t^3 - 3t^2) / [(t^3 + 1)]^2
Combining like terms, we have:
S'(t) = (-8t^3 - 3t^2 + 4) / [(t^3 + 1)]^2
Now, to find how fast the sales are changing one year after the laser discs are released, we evaluate S'(t) at t = 1.
Substituting t = 1 into the equation for S'(t), we get:
S'(1) = (-8(1)^3 - 3(1)^2 + 4) / [(1^3 + 1)]^2
= (-8 - 3 + 4) / [2]^2
= (-7) / 4
= -7/4
Therefore, the sales are changing at a rate of -7/4 million dollars per year one year after the laser discs are released.