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.

Explanation:

• For \( f(2) = 3 \), in the context of the function \( f(t) \) where \( t \) is the time after injection and \( f(t) \) is the amount of drug in the bloodstream. When \( t = 2 \) (2 hours after injection), the value of the function \( f(2)=3 \) means the amount of drug in the bloodstream is 3mg.
• For \( f^{\prime}(2)=-0.3 \), the derivative \( f^{\prime}(t) \) represents the rate of change of the function \( f(t) \). A negative derivative value indicates that the function is decreasing. So when \( t = 2 \), \( f^{\prime}(2)=-0.3 \) means the amount of drug in the bloodstream is falling at a rate of \( 0.3 \)mg per hour.

Answer:

• For “What is the meaning of \( f(2) = 3 \)?”: A. 2 hours after the drug was injected, the amount present in the bloodstream is 3mg.
• For “What is the meaning of \( f^{\prime}(2)=-0.3 \)?”: 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.