Question

the function ( t = f(a) ) models the time, in minutes, for a chemical reaction to occur as a function of the amount ( a ) of catalyst used, measured in milliliters. what are the units for ( f(a) )?
a minutes per milliliter
b milliliters per minute
c minutes per milliliter per milliliter
d milliliters per minute per minute

Explanation:

Step1: Recall the units of the first - derivative

The first - derivative \(f^{\prime}(A)=\frac{df}{dA}\). Since \(t = f(A)\) where \(t\) (the dependent variable) has units of minutes and \(A\) (the independent variable) has units of milliliters, the units of \(f^{\prime}(A)\) are \(\frac{\text{minutes}}{\text{milliliters}}\) (minutes per milliliter).

Step2: Find the units of the second - derivative

The second - derivative \(f^{\prime\prime}(A)=\frac{d}{dA}(f^{\prime}(A))\). Let \(y = f^{\prime}(A)\) with \(y\) having units \(\frac{\text{minutes}}{\text{milliliters}}\) and the variable of differentiation is still \(A\) (units of milliliters). Then \(f^{\prime\prime}(A)=\frac{dy}{dA}\), and the units of \(f^{\prime\prime}(A)\) are \(\frac{\frac{\text{minutes}}{\text{milliliters}}}{\text{milliliters}}=\frac{\text{minutes}}{\text{milliliters}\times\text{milliliters}}\) (minutes per milliliter per milliliter).

Answer:

C. minutes per milliliter per milliliter