The tangent at s = 4 is steeper than the tangent at s = 10. Which statement is correct?

The file arrives at a higher rate at 4 seconds than at 10 seconds.

More of the file has arrived at 4 seconds than at 10 seconds.

The rate at 4 seconds is 48 megabytes.

The file arrives at a higher rate at 10 seconds, because more has arrived by then.

Topic 4.1

Give the derivative's units

Builds on interpreting a derivative’s value and sign. 7 minutes.

A reservoir fills

A reservoir holds \(V\) cubic meters of water \(t\) hours after a storm begins.

The orange tangent touches the curve at 10 hours. The triangle under it has a faint run to the right and an orange rise up to the tangent.

The rise is a change in volume, so it is in cubic meters. The run is a change in time, so it is in hours.

With a run of 5 hours, the rise is 450 cubic meters, so the slope is \(450 \div 5 = 90\) cubic meters per hour. Change the run. The rise changes with it, but the slope stays 90 because the tangent does not move.

Output per input

Without numbers, the vertical axis counts the output and the horizontal axis counts the input.

Drag the point. Wherever it is, the rise is a change in the output and the run is a change in the input.

The slope is therefore in output units per input unit, and these units say what the derivative measures.

Notation

The derivative is a slope, and a slope is a rise divided by a run.
\[\dfrac{dV}{dt} \approx \dfrac{\cy{\Delta V}}{\Delta t}\]
The rise \(\cy{\Delta V}\) is in cubic meters, the units of the output. The run \(\Delta t\) is in hours, the units of the input.
The units of the derivative are therefore output units per input unit.
\[\dfrac{\text{m}^3}{\text{hour}}\]
Write the units beside the number, because they say what the number measures.
\[V'(10) = 90\ \text{m}^3\text{ per hour}\]
To change units, convert the top and the bottom separately. One cubic meter is 1000 liters, and one hour is 60 minutes.
\[\begin{aligned} 90\,\dfrac{\text{m}^3}{\text{h}} &= \dfrac{90 \cdot 1000\ \text{L}}{60\ \text{min}} \\ &= 1500\,\dfrac{\text{L}}{\text{min}} \end{aligned}\]

When it fails

The curve and the tangent do not move, and only the numbers on the time axis change. A run of 1 minute is one-sixtieth of an hour, so its rise is one-sixtieth of 90.

The slope is now 1.5 cubic meters per minute. The same rate has a different number when the units change, so a number needs its units.

Example 1: Air temperature

The air temperature is \(T(a) = 15 - 6.5a\) degrees Celsius at an altitude of \(a\) kilometers. Find the units of \(T'(a)\) and its value.

The output is the temperature, in degrees Celsius, and the input is the altitude, in kilometers.
This means that \(T'\) is in degrees Celsius per kilometer.
\[\dfrac{^\circ\text{C}}{\text{km}}\]
The graph is a line with slope \(-6.5\), so \(T'(a) = -6.5\) at every altitude. The rise is red because it is negative: the air cools 6.5 degrees Celsius for each kilometer up.
\[T'(a) = \cr{-6.5}\]

Example 2: A packing line

A packing line has boxed \(N(t) = 300t + 90t^2\) boxes after \(t\) hours. Find \(N'(3)\) in boxes per minute.

Differentiate \(N\), and substitute 3 into \(N'\). The output is boxes and the input is hours, so the units are boxes per hour.
\[\begin{aligned} N'(t) &= 300 + 180t \\ N'(3) &= \cy{840}\ \text{boxes per hour} \end{aligned}\]
A tempting wrong move: To change hours to minutes, multiply by 60.
\[\begin{gathered} 840 \times 60 = 50\,400 \\ \text{boxes per minute} \end{gathered}\]
The triangle for a run of 1 minute is tiny, because in 1 minute the line packs one-sixtieth of what it packs in 1 hour. The rate per minute must be less than 840, not more.
The hour is in the bottom of the ratio, and an hour has 60 minutes, so divide by 60.
\[\dfrac{840}{60} = \cy{14}\ \text{boxes per minute}\]

Practice

A taxi charges \(C(d) = 3 + 2.4d\) dollars for a ride of \(d\) kilometers. Find the units of \(C'(d)\). Then write its value in cents per meter.

The rise is in dollars and the run is in kilometers. Convert the dollars to cents and the kilometers to meters separately.

Solution

The graph of \(C\) is a line with slope 2.4. The output is dollars and the input is kilometers, so \(C'\) is in dollars per kilometer.
\[C'(d) = \cy{2.4}\ \text{dollars per kilometer}\]
One dollar is 100 cents, and one kilometer is 1000 meters. Convert the top and the bottom separately.
\[\dfrac{2.4 \times 100\ \text{cents}}{1000\ \text{m}}\]
A ride costs 0.24 cents more for each additional meter.
\[\dfrac{240}{1000} = \cy{0.24}\ \text{cents per meter}\]

Questions

These questions use the number of megabytes \(D\) downloaded after \(s\) seconds. The vertical axis counts megabytes, and the horizontal axis counts seconds.

The point is at \(s = 10\). What are the units of the slope of the tangent?
- Megabytes per second
- Megabytes
- Seconds per megabyte
- Megabyte-seconds
Answer
Megabytes per second
The tangent at \(s = 4\) is steeper than the tangent at \(s = 10\). Which statement is correct?
- The file arrives at a higher rate at 4 seconds than at 10 seconds.
- More of the file has arrived at 4 seconds than at 10 seconds.
- The rate at 4 seconds is 48 megabytes.
- The file arrives at a higher rate at 10 seconds, because more has arrived by then.
Answer
The file arrives at a higher rate at 4 seconds than at 10 seconds.
A 3D printer uses material at a rate of 0.6 grams per minute. How many kilograms per hour is this?
- 0.036 kilograms per hour
- 36 kilograms per hour
- 0.01 kilograms per hour
- 0.6 kilograms per hour
Answer
0.036 kilograms per hour

Summary

A derivative is a slope, a rise in output units over a run in input units, so its units are output units per input unit.

You write these units beside the number, because they say what the rate measures.

When a question asks for other units, you convert the top and the bottom of the ratio separately.

This is one of four skills in topic 4.1 of Which & Why: AP Calculus.