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 \(f^{\prime}(A)=\frac{dt}{dA}\). The unit of \(t\) (time) is minutes and the unit of \(A\) (amount of catalyst) is milliliters. So, the unit of \(f^{\prime}(A)\) is \(\frac{\text{minutes}}{\text{milliliter}}\).

Step2: Recall the formula for 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 \(\frac{\text{minutes}}{\text{milliliter}}\) and the variable of differentiation is \(A\) (units: milliliters). Then \(f^{\prime\prime}(A)=\frac{dy}{dA}\).
Using the quotient rule for units: if \(y=\frac{u}{v}\) and we differentiate with respect to \(x\), then \(\frac{dy}{dx}\) has units \(\frac{\frac{u}{v}}{x}\). Here, \(u\) (unit: minutes), \(v\) (unit: milliliter), \(x\) (unit: milliliter). So the unit of \(f^{\prime\prime}(A)\) is \(\frac{\frac{\text{minutes}}{\text{milliliter}}}{\text{milliliter}}=\text{minutes per milliliter per milliliter}\)

Answer:

C. minutes per milliliter per milliliter