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Activity3.4Derivative Units

We can think about the instantaneous velocity as being the instantaneous rate of change in position. In general, whenever you see the phrase “rate of change” you can assume that the rate of change at one instant is being discussed. When we want to discuss average rates of change over a time interval we always say “average rate of change.”

In general, if \(f\) is any function, then \(\fe{\fd{f}}{a}\) tells us the rate of change in \(f\) at \(a\). Additionally, if \(f\) is an applied function with an input unit of \(i_{\text{unit}}\) and an output unit of \(f_{\text{unit}}\text{,}\) then the unit on \(\fe{\fd{f}}{a}\) is \(\frac{f_{\text{unit}}}{i_{\text{unit}}}\text{.}\) Please note that this unit loses all meaning if it is simplified in any way. Consequently, we do not simplify derivative units in any way, shape, or form.

For example, if \(\fe{v}{t}\) is the velocity of your car (measured in mi⁄h) where \(t\) is the amount of time that has passed since you hit the road (measured in minutes), then the unit on \(\fe{\fd{v}}{t}\) is \(\frac{\text{mi}/\text{h}}{\text{min}}\text{.}\)

Subsection3.4.1Exercises

Determine the unit for the first derivative function for each of the following functions. Remember, we do not simplify derivative units in any way, shape, or form.

1

\(\fe{V}{r}\) is the volume of a sphere (measured in mL) with radius \(r\) (measured in mm).

2

\(\fe{A}{x}\) is the area of a square (measured in ft2) with sides of length \(x\) (measured in ft).

3

\(\fe{V}{t}\) is the volume of water in a bathtub (measured in gal) where \(t\) is the amount of time that has elapsed since the bathtub began to drain (measured in minutes).

4

\(\fe{R}{t}\) is the rate at which a bathtub is draining (measured in gal⁄min) where \(t\) is the amount of time that has elapsed since the bathtub began to drain (measured in minutes).

5

Akbar was given a formula for the function described in Exercise 3.4.1.3. Akbar did some calculations and decided that the value of \(\fe{\fd{V}}{2}\) was (without unit) \(1.5\text{.}\) Nguyen took one look at Akbar's value and said “that's wrong” What is it about Akbar's value that caused Nguyen to dismiss it as wrong?

6

After a while Nguyen convinced Akbar that he was wrong, so Akbar set about doing the calculation over again. This time Akbar came up with a value of \(-12528\text{.}\) Nguyen took one look at Akbar's value and declared “still wrong.” What's the problem now?

7

What would it mean if the value of \(\fe{R}{t}\) was zero for all \(t\gt0\text{?}\)

8

What would it mean if the value of \(\fe{\fd{R}}{t}\) was zero for all \(t\) in \(\ointerval{0}{2.25}\text{?}\)