- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Exponential and trigonometric functions practice question 17
A bacterial culture contains cells hours after it is started. How fast is the culture growing at ?
Give the answer in cells per hour.
How fast the culture grows is . In the variable sits in the exponent: which derivative rule fits that, and which one does not?
The variable is in the exponent, so this is an exponential, not a power: , where is the natural logarithm.
The constant 500 stays in front. Substitute t = 2 at the end.
An exact value, or a decimal rounded to the nearest cell per hour.
Theory UBC Math 100 §3.5CLP-1 §2.7Key results
Worked solution
Strategy
Differentiate the exponential with base 3, keeping the constant 500, then evaluate at t = 2.
Working
Constant base, variable exponent: aᵗ differentiates to aᵗ log a.
Substitute t = 2: 3² = 9.
Answer
The answer is 4500 times log 3, about 4944 cells per hour.
Check
N gains 3000 cells from t = 1 to 2 and 9000 from t = 2 to 3, so its rate at t = 2 lies between 3000 and 9000 per hour.
Where this goes wrong
Using the power rule, 500 · t · 3t−1, which gives 3000. The power rule needs a constant exponent; here the exponent is the variable t.
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