- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Exponential growth and decay practice question 1
Which of the following functions satisfies the differential equation ?
Differentiate each candidate and compare with three times the candidate. Only one matches for every .
The chain rule gives , and an added constant differentiates to 0. Work out for each choice and set it beside .
Theory UBC Math 100 §11.4CLP-1 §3.3Key results
Worked solution
Strategy
The equation says the rate is three times the amount. Test each candidate by differentiating; only a constant multiple of e3t works.
Working
The chain rule brings down the 3.
Each of the others fails.
Answer
The answer is y equals 5 e to the 3t.
Check
At t = 0, y = 5e3t has value 5 and slope 15, three times the value, while e3t + 5 has value 6 but slope only 3.
Where this goes wrong
Choosing e3t + 5 because it “has e3t in it”. Adding a constant breaks the equation: the derivative ignores the 5 but 3y does not.
Leans on
The derivative of ekt, from The chain rule and logarithms
Try a similar question
A quantity satisfies for all , and . Find .Saved in this browser only.