- Topic
- Antiderivatives
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
- Theory
- CLP-1 §4.1Key results
Antiderivatives practice question 7
A colony of bacteria grows at a rate of individuals per hour, where is measured in hours. By how much does the colony grow during the first 5 hours?
Give the answer to the nearest individual.
The growth is a change in the population, and you are given its rate of change. Which function has that rate as its derivative?
If is the population then . Find up to a constant; the growth over is , and the constant cancels.
The antiderivative of is : dividing by the inner constant undoes the chain rule.
An exact value, or the number rounded to the nearest individual.
Theory CLP-1 §4.1Key results
Worked solution
Strategy
A rate integrates to a total. The antiderivative of the rate, evaluated at the end and the start, gives the change; the unknown constant drops out of the difference.
Working
Antiderivative of the rate; dividing by 0.1 undoes the chain rule.
The constant C cancels.
Answer
The answer is about 1297 individuals.
Check
The rate climbs from 200 to about 330 an hour, so 5 hours bring between 5 × 200 = 1000 and 5 × 330 = 1650 new individuals, and 1297 is in that range.
Where this goes wrong
Multiplying the initial rate by 5 hours to get 1000. The rate grows over the interval, so the total is larger; only the antiderivative accounts for that.
Leans on
The derivative of ekt, from The chain rule and logarithms
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