- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
The definition of the derivative practice question 6
A roastery’s cost of roasting kilograms of coffee in a day is dollars, and . Which statement is the best reading of ?
What units does carry? Test each statement against those units and against what a derivative measures.
is a rate: its units are dollars per kilogram. It says how fast the cost grows as increases past 50, not what 50 kg cost.
Theory UBC Math 100 §3.3CLP-1 §2.3Key results
Worked solution
Strategy
A derivative is a rate of change. C′(50) is measured in dollars per kilogram, and it estimates what one more kilogram adds to the cost when the roastery is at 50 kg a day.
Working
The numerator is in dollars and the denominator in kilograms, so the units are dollars per kilogram.
Over a step of h = 1 kg the change in cost is about the rate times the step: the next kilogram adds about $12.
Answer
The answer is at 50 kg a day, one more kilogram adds about 12 dollars to the cost.
Check
For C(q) = 300 + 12q, C′(50) = 12 and the 51st kilogram adds exactly $12, while C(50) = $900 and the average cost is $18 a kilogram.
Where this goes wrong
Reading C′(50) as C(50), the cost of 50 kg. The prime makes it a rate: what the next kilogram costs, in dollars per kilogram.
Saved in this browser only.