The Derivative as a Rate of Change · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

The Derivative as a Rate of Change

Mathematics · Calculus & Analysis · ages 17-18
Name ______________________   Date ____________

  1. A runner has covered s metres after t seconds. What real quantity does ds/dt measure?

    • the total distance run
    • the time taken so far
    • the runner's pace in seconds for each metre
    • the runner's speed in metres per second

  2. A derivative of zero means the quantity itself is zero.

    Circle one:   True   False

  3. C is a cost in pounds and n is a number of items. What are the units of dC/dn?

    • pounds
    • pounds per item
    • items per pound
    • pounds times items

  4. A seedling's mass M is measured in grams and the time t in days. What are the units of dM/dt?

    • Days per gram
    • Grams per day
    • Grams times days
    • Grams

  5. The water depth D in a harbour has dD/dt = 0 at noon. What does that mean?

    • The depth is momentarily steady, neither rising nor falling
    • The harbour is empty
    • The depth is falling fast
    • The tide has stopped forever and will never move again afterwards

  6. If dV/dt = 0 at one instant, the volume will never change again.

    Circle one:   True   False

  7. Water in a tank is modelled by V(t), with V in litres and t in seconds. At one moment dV/dt = -3. What does that mean?

    • the tank is losing 3 litres every second
    • the tank holds 3 litres
    • the tank is gaining 3 litres every second
    • the tank will be empty in 3 seconds

  8. A firm's revenue is R = 30n - 0.05n² pounds from n items. What is dR/dn at n = 100?

    • 25 pounds per item
    • 20 pounds per item
    • 30 pounds per item
    • 2500 pounds per item

  9. A workshop's cost is C = 1200 + 5n + 0.02n² pounds for n items. What is dC/dn at n = 50, in pounds per item?

    Answer: ______________

  10. Nina works out dV/dt = -4 for a tank and reports that it is filling at 4 litres per second. Where is the slip?

    • the units should have been seconds, not litres per second
    • the answer should have been -4t rather than -4
    • a rate cannot be negative, so the differentiation must be wrong
    • a negative rate means the volume is falling, so it is emptying
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Answer key

For grown-ups. Fold this page away before handing over the rest.

The Derivative as a Rate of Change W1-mt_RqyLE3jrAW-s1

  1. the runner's speed in metres per second · Distance per unit of time is speed. That is what a derivative is: how much of the top quantity you get per unit of the bottom one.
  2. False · Zero rate means the quantity is not changing at that instant. A tank holding a steady 200 litres has a derivative of zero while holding 200 litres, not 0.
  3. pounds per item · Pounds sit on the top and items on the bottom, so the units are pounds per item: what one more item adds to the bill.
  4. Grams per day · A derivative divides the output units by the input units, so grams divided by days means grams per day.
  5. The depth is momentarily steady, neither rising nor falling · A zero rate means no change at that instant, so the depth is momentarily steady.
  6. False · A single instant of zero says only that the volume is momentarily still. It can rise or fall immediately afterwards, and usually does.
  7. the tank is losing 3 litres every second · The size 3 gives the speed and the units make it litres per second. The minus sign says the volume is falling.
  8. 20 pounds per item · dR/dn = 30 - 0.1n. At n = 100 that is 30 - 10, which is 20 pounds per item.
  9. 7 · dC/dn = 5 + 0.04n. At n = 50 that is 5 + 2, which is 7 pounds per item.
  10. a negative rate means the volume is falling, so it is emptying · She read the size and dropped the sign. Negative means the quantity goes down as time goes up, so the tank is losing 4 litres a second.
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