- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Slope fields and Euler’s method practice question 8
Early in the season an insect population grows at a per-capita rate that falls as the season goes on. With in thousands of insects and in weeks,
Use Euler’s method with two steps of to estimate .
Here the slope changes with time as well as with the population. At the start of each step, which two values do you need before you can work it out?
The slope depends on both and : . Each step is .
The first step uses ; the second uses , with .
A number.
Theory UBC Math 100 §12.3Key results
Worked solution
Strategy
Step along the tangent twice. Each step uses the slope at the point it starts from, which needs both the current time and the current population.
Working
First step, from t = 0.
Second step, from t = 1: both t and P have moved on.
For comparison: differentiating shows this formula solves the problem, and the two-step estimate is close to it.
Answer
The answer is about 224 thousand insects.
Check
Over two weeks the per-capita rate averages 0.4, so P grows by a factor of about e0.8 ≈ 2.23, to roughly 223 thousand, close to 224.
Where this goes wrong
Leaving t = 0 in the slope for the second step, which gives 256. By t = 1 the per-capita rate has fallen to 0.4, and the second step must use it.
Leans on
Linear approximation, from Linear approximation
Saved in this browser only.