- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
The definition of the derivative practice question 3
Let be the temperature of a lake, in degrees Celsius, at a depth of metres below the surface on a summer afternoon. What are the units of , and what sign would you expect it to have?
A derivative has the units of the output divided by the units of the input. Its sign says whether the output rises or falls as the input increases.
The difference quotient divides a change in temperature by a change in depth. On a summer afternoon, is the water near the surface warmer or colder than the water further down?
Theory UBC Math 100 §3.3CLP-1 §2.3Key results
Worked solution
Strategy
Read the derivative as a rate: how much the output changes per unit of input. Units follow from that, and the sign follows from which direction the output moves.
Working
Numerator in °C, denominator in metres.
Going one metre deeper in a sun-warmed lake takes you to colder water, so T decreases as d increases and T′(d) is negative.
Answer
The answer is degrees Celsius per metre, and negative.
Check
A lake at 22 °C on top and 14 °C 4 m down changes at (14 − 22)/4 = −2 °C per metre on average: those units, and negative.
Where this goes wrong
Reasoning about the sign of T rather than the sign of the change in T. The temperature is positive; the rate at which it changes with depth is what the derivative measures, and that is negative.
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