Question

suppose that 5 mg of a drug is injected into the bloodstream. let f(t) be the amount present in the bloodstream after t hours. interpret f(2)=3 and f(2)= - 0.3. estimate the number of milligrams of the drug in the bloodstream after 2 1/4 hours.
what is the meaning of f(2)=3?
a. 2 hours after the drug was injected, the amount present in the bloodstream is 3 mg.
b. 2 hours after the drug was injected, the amount present in the bloodstream is rising at a rate of 3 mg per hour.
c. 3 hours after the drug was injected, the amount present in the bloodstream is rising at a rate of 2 mg per hour.
d. 3 hours after the drug was injected, the amount present in the bloodstream is 2 mg.
what is the meaning of f(2)= - 0.3?
a. 2 hours after the drug was injected, the amount present in the bloodstream is falling at a rate of 0.3 mg per hour.
b. 2 hours after the drug was injected, the amount present in the bloodstream is rising at a rate of 0.3 mg per hour.
c. 0.3 hour after the drug was injected, the amount present in the bloodstream is 2 mg.
d. 2 hours after the drug was injected, the amount present in the bloodstream is - 0.3 mg.
after 2 1/4 hours, the number of milligrams of the drug in the bloodstream will be □ mg.
(type an integer or a decimal.)

Explanation:

Step1: Use the linear approximation formula

The linear approximation formula is \(L(t)=f(a)+f^{\prime}(a)(t - a)\). Here, \(a = 2\), \(t=2\frac{1}{4}=2.25\), \(f(2) = 3\), and \(f^{\prime}(2)=- 0.3\).

Step2: Substitute the values into the formula

Substitute \(a = 2\), \(t = 2.25\), \(f(2)=3\), and \(f^{\prime}(2)=-0.3\) into \(L(t)=f(a)+f^{\prime}(a)(t - a)\).
We get \(L(2.25)=f(2)+f^{\prime}(2)(2.25 - 2)\).

Step3: Calculate the result

First, calculate \((2.25 - 2)=0.25\). Then, \(f^{\prime}(2)(2.25 - 2)=-0.3\times0.25=- 0.075\).
Since \(f(2) = 3\), then \(L(2.25)=3-0.075 = 2.925\).

Answer:

\(2.925\)