Unit 4 · Topic 4.3 Beta

Rates of Change in Other Contexts

In any context, Q′ is the rate of change of Q in units of Q per input unit, and Q″ is the rate of change of that rate.

Practice 3: JustificationPractice 4: Communication and Notation

Question set for this topic

Part 1 · Hook

Why this matters

Doctors track how fast a medicine's concentration falls, economists how fast cost grows with each extra unit, ecologists how fast a population spreads. The derivative is the same in all of them; what changes is the story and the units. This topic practices reading rates, and rates of rates, in many settings.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. The units of f′(t)f'(t), when ff is in liters and tt in minutes, are

  1. liters per minute
  2. liters
  3. minutes per liter
  4. liters times minutes
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Output units per input unit.

  • Correct: liters per minute:
  • liters:
  • minutes per liter:
  • liters times minutes:

2. For a particle, v(3)=−2v(3) = -2 and a(3)=−1a(3) = -1. At t=3t = 3 the particle is

  1. speeding up
  2. slowing down
  3. at rest
  4. changing direction
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vv and aa have the same sign.

  • Correct: speeding up:
  • slowing down:
  • at rest:
  • changing direction:

Part 4 · See it

See it first

Top: temperature H(t) = 70 − 50e^(−0.1t) of a drink in °F for t from 0 to 40 minutes, about 51.6 °F at t = 10. Bottom: its rate H′(t) = 5e^(−0.1t) in °F per minute, about 1.84 at t = 10 and shrinking.
A drink's temperature H(t)H(t) in °F (top) and its rate of change H′(t)H'(t) in °F per minute (bottom). At 10 minutes, H′(10)≈1.84H'(10) \approx 1.84 °F per minute, the slope of the tangent line above. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Any quantity QQ that depends on an inputhas a rate of change Q′Q' in units of QQ per unit of input
  2. The sign of Q′Q'says whether the quantity is increasing or decreasing
  3. Q′Q' is itself a function, so Q″Q'' is the rate of change of the ratemeasured in units of Q per input unit per input unit
  4. The sign of Q″Q''says whether the rate Q′Q' is increasing or decreasing

Part 6 · Key ideas

Key ideas

  • An interpretation names the input value, the quantity, increasing or decreasing, and the rate with units.
  • When the function is itself a rate (liters per hour), its derivative is a rate of a rate (liters per hour per hour).
  • Q′<0Q' < 0 with Q″>0Q'' > 0: decreasing, but more slowly. Q′>0Q' > 0 with Q″<0Q'' < 0: increasing, but more slowly.
  • Chained quantities multiply rates, and their units cancel: (°F per foot)(feet per minute) = °F per minute.

Part 7 · Misconception

A common mistake

The wrong idea: If H″(t)>0H''(t) > 0, the temperature is going up.

What actually happens: H″H'' describes the rate H′H', not the temperature. With H′<0H' < 0 and H″>0H'' > 0 the temperature is still falling, just more slowly.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

A bowl of soup cooling: H(t)=20+70e−0.04tH(t) = 20 + 70e^{-0.04t}

1020304050600102030405060708090100t (minutes)H (°C)H

1. The temperature of a bowl of soup is H(t)=20+70e−0.04tH(t) = 20 + 70e^{-0.04t} degrees Celsius, tt minutes after it is served. Find H′(10)H'(10), with units.

Calculator allowed. Give the answer correct to three decimal places, then its unit.

Show the answer

H′(t)H'(t)=70(−0.04)e−0.04t= 70(-0.04)e^{-0.04t}=−2.8e−0.04t= -2.8e^{-0.04t}, so H′(10)H'(10)=−2.8e−0.4= -2.8e^{-0.4}≈−1.877\approx -1.877 °C per minute. (A calculator's numerical derivative at t=10t = 10 gives the same value.)

  • Answer: -1.877 degrees Celsius per minute

2. What are the units of H″(t)H''(t)?

  1. degrees Celsius per minute per minute
  2. degrees Celsius per minute
  3. degrees Celsius
  4. minutes per degree Celsius
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Each derivative with respect to tt divides by minutes once more: H″(t)H''(t) is in degrees Celsius per minute per minute.

  • Correct: degrees Celsius per minute per minute: H″H'' is the rate of change of H′H', which is in °C per minute.
  • degrees Celsius per minute: Those are the units of H′(t)H'(t).
  • degrees Celsius: Those are the units of H(t)H(t).
  • minutes per degree Celsius: Rates put the output on top: °C per minute, then per minute again.

3. Find H″(10)H''(10), with units.

Calculator allowed. Give the answer correct to three decimal places, then its unit.

Show the answer

H″(t)H''(t)=(−2.8)(−0.04)e−0.04t= (-2.8)(-0.04)e^{-0.04t}=0.112e−0.04t= 0.112e^{-0.04t}, so H″(10)H''(10)=0.112e−0.4= 0.112e^{-0.4}≈0.075\approx 0.075 °C per minute per minute.

  • Answer: 0.075 degrees Celsius per minute per minute

4. At t=10t = 10, H′(10)<0H'(10) < 0 and H″(10)>0H''(10) > 0. What does this say about the soup?

  1. It is warming, because H″(10)H''(10) is positive
  2. It is cooling, and its rate of cooling is slowing down
  3. It is cooling faster and faster
  4. Its temperature is about to stop falling and start rising
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H′(10)<0H'(10) < 0 means the temperature is decreasing at t=10t = 10. H″(10)>0H''(10) > 0 means H′H' is increasing, toward 0: the soup cools more and more slowly.

  • It is warming, because H″(10)H''(10) is positive: The sign of H′H', not of H″H'', says whether the temperature rises or falls.
  • Correct: It is cooling, and its rate of cooling is slowing down: H′<0H' < 0: cooling. H″>0H'' > 0: H′H' is rising toward 0, so the cooling gets gentler.
  • It is cooling faster and faster: Cooling faster would need H′H' to become more negative, that is H″<0H'' < 0.
  • Its temperature is about to stop falling and start rising: H′(t)=−2.8e−0.04tH'(t) = -2.8e^{-0.04t} is negative for every tt: the soup keeps cooling, more and more slowly.

5. W(t)W(t) is the number of gallons of water in a pool tt hours after filling starts. What are the units of W″(t)W''(t)?

  1. gallons per hour per hour
  2. gallons per hour
  3. gallons squared per hour
  4. hours per gallon per gallon
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W″(t)=ddtW′(t)W''(t) = \frac{d}{dt}W'(t): (gallons per hour) per hour.

  • Correct: gallons per hour per hour: W′W' is in gallons per hour; its rate of change adds another per hour.
  • gallons per hour: Those are the units of W′(t)W'(t).
  • gallons squared per hour: Differentiating twice divides by hours twice; it does not square the gallons.
  • hours per gallon per gallon: Rates put the output on top.

6. The area A(t)A(t) of an oil slick satisfies A′(t)>0A'(t) > 0 and A″(t)<0A''(t) < 0 on an interval. What is happening to the slick?

  1. It is shrinking, but more and more slowly
  2. It is spreading faster and faster
  3. It is spreading, but more and more slowly
  4. It is shrinking faster and faster
Show the answer

The sign of A′A' says the area is growing; the sign of A″A'' says that growth rate is falling.

  • It is shrinking, but more and more slowly: A positive A′A' means the area is growing.
  • It is spreading faster and faster: That would need A″>0A'' > 0.
  • Correct: It is spreading, but more and more slowly: A′>0A' > 0: area increasing. A″<0A'' < 0: the rate of increase is decreasing.
  • It is shrinking faster and faster: With A′>0A' > 0 the slick is not shrinking.

Part 9 · Summary

Summary

In any context, Q′Q' is the rate of change of QQ in units of QQ per input unit, and Q″Q'' is the rate of change of that rate. Read the sign of Q′Q' for increasing or decreasing and the sign of Q″Q'' for whether that rate grows or shrinks. Always state units.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections