Suppose that the proportion P of voters who recognize a candidate's name t months after the start of the campaign is given by the following equation. (Round your answers to three decimal places.)
P(t) =
11t
t2 + 100
+ 0.14
(a) Find the rate of change of P when t = 6.
(b) Find the rate of change of P when t = 12
To find the rate of change of P, we need to differentiate the equation P(t) with respect to t.
(a) To find the rate of change of P when t = 6, we will find the derivative of P(t) with respect to t and then substitute t = 6 into the derivative.
P(t) = (11t / (t^2 + 100)) + 0.14.
To differentiate P(t), we need to use the quotient rule:
P'(t) = [(11(t^2 + 100) - 11t(2t)) / (t^2 + 100)^2].
Now, substitute t = 6 into P'(t):
P'(6) = [(11(6^2 + 100) - 11(6)(2(6))) / (6^2 + 100)^2].
P'(6) = [(11(36 + 100) - 11(6)(12)) / (36 + 100)^2].
P'(6) = [(11(136) - 11(6)(12)) / (136)^2].
P'(6) = [(1496 - 792) / (136)^2].
P'(6) = [704 / (136)^2].
P'(6) ≈ 0.029.
Therefore, the rate of change of P when t = 6 is approximately 0.029.
(b) To find the rate of change of P when t = 12, we will repeat the same steps as above, but substitute t = 12 into the derivative.
P(t) = (11t / (t^2 + 100)) + 0.14.
P'(t) = [(11(t^2 + 100) - 11t(2t))/ (t^2 + 100)^2].
P'(12) = [(11(12^2 + 100) - 11(12)(2(12)))/ (12^2 + 100)^2].
P'(12) = [(11(144 + 100) - 11(12)(24))/ (144 + 100)^2].
P'(12) = [(11(244) - 11(12)(24))/ (244)^2].
P'(12) = [(2684 - 3168) / (244)^2].
P'(12) = [-484 / (244)^2].
P'(12) ≈ -0.008.
Therefore, the rate of change of P when t = 12 is approximately -0.008.
To find the rate of change of P at a specific value of t, we need to find the derivative of the function P(t) with respect to t.
(a) To find the rate of change of P when t = 6, we need to calculate P'(6) or the derivative of P(t) evaluated at t = 6.
First, let's find the derivative of P(t). We can use the quotient rule for differentiation to differentiate the function P(t).
P'(t) = (d/dt)(11t/(t^2 + 100)) + (d/dt)(0.14)
To differentiate the first term, we can use the quotient rule:
(d/dt)(11t/(t^2 + 100)) = (11(t^2 + 100) - 11t(2t))/(t^2 + 100)^2
Simplifying this expression:
(11t^2 + 1100 - 22t^2)/(t^2 + 100)^2 = (-11t^2 + 1100)/(t^2 + 100)^2
So, P'(t) = (-11t^2 + 1100)/(t^2 + 100)^2 + 0.14
To find the rate of change of P when t = 6, we substitute t = 6 into P'(t):
P'(6) = (-11(6)^2 + 1100)/((6)^2 + 100)^2 + 0.14
Simplifying further, we get:
P'(6) = (-396 + 1100)/(36 + 100)^2 + 0.14
P'(6) = 704/136^2 + 0.14
P'(6) ≈ 0.019
Therefore, the rate of change of P when t = 6 is approximately 0.019.
(b) To find the rate of change of P when t = 12, we follow a similar process.
First, let's find the derivative of P(t) using the quotient rule:
P'(t) = (-11t^2 + 1100)/(t^2 + 100)^2 + 0.14
To find P'(12), we substitute t = 12 into P'(t):
P'(12) = (-11(12)^2 + 1100)/((12)^2 + 100)^2 + 0.14
Simplifying further:
P'(12) = (-1584 + 1100)/(144 + 100)^2 + 0.14
P'(12) = -484/244^2 + 0.14
P'(12) ≈ 0.045
Therefore, the rate of change of P when t = 12 is approximately 0.045.