The concentration C (in milligrams per milliliter) of a drug in a patient's bloodstream t hours after injection into muscle tissue is modeled by
C =
3t/26 + t^3
Use differentials to approximate the change in the concentration when t changes from
t = 1
to
t = 1.5.
(Round your answer to four decimal places.)
Well, well, well... let's see what we can do here!
First, let's find the derivative of the concentration function C(t) with respect to t:
dC/dt = 3/26 + 3t^2
Now, let's plug in the values t = 1 and t = 1.5 into this derivative to find the rate of change at those points:
dC/dt (at t = 1) = 3/26 + 3(1)^2 = 3/26 + 3 = 3/26 + 78/26 = 81/26
dC/dt (at t = 1.5) = 3/26 + 3(1.5)^2 = 3/26 + 3(2.25) = 3/26 + 6.75 = 3/26 + 177/26 = 180/26
Now, we want to approximate the change in concentration when t changes from 1 to 1.5. This is approximately equal to the change in C over the change in t, so:
Change in C = dC/dt * Change in t
Change in t = 1.5 - 1 = 0.5
Change in C = (81/26) * 0.5 = 81/52 ≈ 1.5577
So, when t changes from 1 to 1.5, the concentration of the drug in the patient's bloodstream is estimated to change by approximately 1.5577 milligrams per milliliter.
Remember, these are just approximations. Don't take them too seriously!
To approximate the change in concentration using differentials, we need to find the derivative of the concentration function with respect to t.
The given concentration function is:
C = (3t/26) + t^3
Let's differentiate this function with respect to t.
dC/dt = (3/26) + 3t^2
Now we can use this derivative to approximate the change in concentration when t changes from 1 to 1.5.
ΔC = dC/dt * Δt
where Δt = 1.5 - 1 = 0.5
Plugging in the values:
ΔC = (3/26 + 3(1)^2) * 0.5
Simplifying:
ΔC = (3/26 + 3) * 0.5
ΔC = (81/26) * 0.5
ΔC ≈ 1.9615
Therefore, the change in concentration when t changes from t = 1 to t = 1.5 is approximately 1.9615 milligrams per milliliter.
To approximate the change in concentration when t changes from 1 to 1.5, we need to find the derivative of the concentration function with respect to t and evaluate it at t=1.
Given that the concentration function is:
C = (3t/26) + t^3
To find the derivative of this function, we can differentiate each term separately and then add them together.
1. Differentiating the first term:
d/dt (3t/26) = 3/26
2. Differentiating the second term:
d/dt (t^3) = 3t^2
Now we can find the derivative of the concentration function by adding these derivatives together:
dC/dt = (3/26) + 3t^2
To approximate the change in concentration when t changes from 1 to 1.5, we can substitute t=1 into the derivative expression:
dC/dt = (3/26) + 3(1)^2 = 3/26 + 3 = 3/26 + 78/26 = 81/26
This gives us the rate of change of concentration at t=1. Now we can find the approximate change in concentration by multiplying the derivative by the change in t (1.5 - 1):
ΔC ≈ (81/26) * (1.5 - 1)
Calculating this expression will give us the approximate change in concentration when t changes from 1 to 1.5. Let's evaluate it:
ΔC ≈ (81/26) * (1.5 - 1) = (81/26) * 0.5 = 81/52 ≈ 1.5577
Therefore, the approximate change in concentration when t changes from 1 to 1.5 is approximately 1.5577 milligrams per milliliter.
C = 3t/26 + t^3
dC = (3/26 + 3t^2) dt
Just plug in t=1, dt=0.5