# Did I do this function rate of change question Calculus ?

Suppose the function

f(t)= 15t/1+0.25t^2

models a drug in milligrams entering the blood stream at time t in seconds from 0 to 25 seconds.

What is the rate the drug is entering the body at 7 seconds?

I plugged in 7 for t and got 7.9245

How would I find the amount of drug in milligrams that is in the body at 7 seconds?

### 3 Answers

- SpacemanLv 74 weeks ago
You don't even need to use calculus. Just do a spreadsheet

that automatically plugs in the values for each successive t

and f(t) is the amount of the drug in the bloodstream in milligrams.

f(t) = 15t / (1 + 0.25t²)

Time (t) Drug f(t)

seconds milligrams

0 0.0000

1 12.0000

2 15.0000

3 13.8462

4 12.0000

5 10.3448

6 9.0000

7 7.9245

8 7.0588

9 6.3529

10 5.7692

11 5.2800

12 4.8649

13 4.5087

14 4.2000

15 3.9301

16 3.6923

17 3.4812

18 3.2927

19 3.1233

20 2.9703

21 2.8315

22 2.7049

23 2.5891

24 2.4828

25 2.3847

- RealProLv 74 weeks ago
I suggest before doing calculus, you focus on distinguishing between 15t/(1+0.25t^2) and 15t/1+0.25t^2.

Nothing here makes sense. "The function models a drug"? What about the drug is it modeling?

If the statement were

The function models the rate, in mg/s, at which a drug is entering the system at time t in seconds, between 0 and 25 s.

Then the answer to the first question is indeed about 7.92 mg/s.

To find the amount of drug that has entered the system from 0 to 7 s, we would integrate f(t) between t=0 and t=7.

77.52 mg

- AshLv 74 weeks ago
f(t)= 15t/(1+0.25t²)

Amount of drug at t seconds would be the area under the curve from t=0 to t=7

₀∫⁷f(t) dt =₀∫⁷ 15t/(1+0.25t²) dt

Let u = (1+0.25t²)

du = 0.5t dt

30 du = 15t dt

I will not change the limits to u as we will change back to t and then solve. However, you can alternatively change the limit to u and get the same answer.

∫ (30/u) du

= 30∫ (1/u) du

= 30 ln u + C

Now plug back t

= [30 ln(1+0.25t²)] ₀|⁷

= 30 [ln(1 + 0.25(7²) - ln (1 + 0.25(0²))

= 30[ln(13.25) - ln(1)]

= 30[ln(13.25) - 0]

= 30[ln(13.25)]

= 77.5 mg