Wemazi

Topic 4.3

Interpret a signed rate with units

Builds on calculating a derivative in context. 7 minutes.

A reservoir after a storm

A reservoir holds \(V\) thousand cubic meters of water, \(t\) days after a storm. The curve shows \(V\), green where \(V\) rises and red where \(V\) falls.

Drag the orange point. The slope of the orange tangent is the rate \(V'\), which is positive where the curve rises and negative where it falls.

Go to day 15. The slope is \(-6\), so at that instant the water is decreasing at 6 thousand cubic meters per day.

A full reading names four facts: the quantity, the instant, the direction, and the size with units.

What a signed rate says

Without the numbers, the output is y and the input is x. Drag the orange point, and the tangent at that input shows the rate.

If the slope is positive, y increases as x increases. If the slope is negative, y decreases as x increases. If the slope is zero, y neither rises nor falls at that input.

The size of the slope is the change in y for each one unit of x. It holds only at the input where the tangent touches the curve.

Notation

  1. Write the value with its units, which are the units of the output per unit of the input.

    \[\begin{aligned} V'(15) &= \cr{-6} \\ &\text{thousand m}^3 \text{ per day} \end{aligned}\]

  2. Name the instant, which is the input where you take the derivative.

    \[t = 15 \text{ days}\]

  3. Read the sign. A positive value means the quantity increases as the input increases, and a negative value means it decreases.

    \[V'(15) < 0\]

  4. Read the size, which is the change in the quantity for each unit of the input.

    \[|V'(15)| = 6\]

  5. Write one sentence with the four parts.

    At 15 days after the storm, the volume of water is decreasing at 6 thousand cubic meters per day.

When it fails

The rate holds only at its instant. If the rate stayed \(-6\) for 10 days, the volume would fall by 60, to the end of the dashed tangent at day 25.

The curve falls by 120 instead, because it becomes steeper, and at day 25 the rate is \(-18\). The red gap of 60 is the error, so use a rate over a longer interval only where the slope changes little.

Example 1: A cup of tea

The temperature of a cup of tea is \(T = 20 + 70e^{-m/10}\) degrees Celsius, \(m\) minutes after it is poured. Interpret \(T'(10)\).

  1. Differentiate. The chain rule multiplies 70 by \(-\frac{1}{10}\), the derivative of the exponent.

    \[T'(m) = -7e^{-m/10}\]

  2. Substitute \(m = 10\). As a check, the value is about \(-2.6\).

    \[T'(10) = -\dfrac{7}{e}\]

  3. The units are degrees Celsius per minute. The sign is negative, so the temperature decreases.

  4. At 10 minutes after it is poured, the tea is cooling at about 2.6 degrees Celsius per minute.

Example 2: Income tax

The tax on an income of \(I\) dollars is \(T = 0.1I + 0.000002I^2\) dollars. Interpret \(T'(35000)\).

  1. Differentiate. Then substitute \(I = 35000\). The tangent has slope 0.24.

    \[\begin{aligned} T'(I) &= 0.1 + 0.000004I \\ T'(35000) &= 0.24 \end{aligned}\]

  2. A tempting wrong move: The person pays 24% of the whole income as tax.

    \[\cr{0.24 \times 35000 = 8400}\]

    The height of the curve at I = 35000 is the tax itself, 5950 dollars, not 8400. The number 0.24 is the slope of the tangent, and it compares extra tax with extra income.

  3. Read 0.24 as a rate in dollars of tax per dollar of income. At an income of 35000 dollars, each extra dollar of income adds about 0.24 dollars of tax.

Practice

A hiker climbs a mountain. The air pressure is \(P = 100 - 12h + 0.5h^2\) kilopascals at an altitude of \(h\) kilometers. Find \(P'(4)\). Then write one sentence that reads it.

Drag the point to \(h = 4\). The tangent there shows the sign of \(P'(4)\), and the units are those of \(P\) per unit of \(h\).

Solution

  1. Differentiate with respect to \(h\).

    \[P'(h) = -12 + h\]

  2. Substitute \(h = 4\).

    \[P'(4) = \cr{-8}\]

  3. The sign is negative, so the pressure decreases as the altitude increases. At an altitude of 4 kilometers, the pressure is decreasing at 8 kilopascals per kilometer.

Questions

The graph shows the temperature \(H\) of a greenhouse, in degrees Celsius, \(t\) hours after midnight. Question 3 does not use the graph.

  1. The tangent at \(t = 12\) has slope \(-2\). Which sentence reads it correctly?

    • At 12 hours after midnight, the temperature is decreasing at about 2 degrees Celsius per hour.
    • At 12 hours after midnight, the temperature is increasing at about 2 degrees Celsius per hour.
    • The temperature falls by 2 degrees in total after 12 hours.
    • At 12 hours after midnight, the temperature is decreasing at about 2 hours per degree.
    Answer

    At 12 hours after midnight, the temperature is decreasing at about 2 degrees Celsius per hour.

  2. How does the rate at \(t = 14\) compare with the rate at \(t = 12\)?

    • The temperature falls faster at \(t = 14\)
    • The temperature falls more slowly at \(t = 14\)
    • The temperature rises at \(t = 14\)
    • The temperature does not change at \(t = 14\)
    Answer

    The temperature falls faster at \(t = 14\)

  3. The depth of water in a tank is \(d = 5 - 0.4t^2\) meters, \(t\) hours after the tank starts to drain. Which sentence reads \(d'(3)\) correctly?

    • At 3 hours, the depth is decreasing at 2.4 meters per hour.
    • At 3 hours, the depth is increasing at 2.4 meters per hour.
    • At 3 hours, the depth is 2.4 meters.
    • At 3 hours, the depth is decreasing at 2.4 hours per meter.
    Answer

    At 3 hours, the depth is decreasing at 2.4 meters per hour.

Summary

You read a signed rate as one sentence with four parts: the quantity, the instant, the direction, and the size with units.

The sign gives the direction, because the slope is positive where the quantity rises and negative where it falls.

A derivative in context is a rate at one instant, not a total change or a share of the whole amount.

This is one of four skills in topic 4.3 of Which & Why: AP Calculus.