- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Slope fields and Euler’s method practice question 3
A loaf of bread comes out of the oven at 95 °C into a kitchen at 25 °C. Its temperature , in °C after minutes, satisfies . Use one step of Euler’s method with to estimate the temperature after 2 minutes.
Euler’s method walks along the tangent line from the starting point. What slope does the equation give at 95 °C, and how many minutes does the step cover?
Euler’s method steps along the tangent: , where is the slope.
A number.
Theory UBC Math 100 §12.3Key results
Worked solution
Strategy
Over a short time the temperature changes at roughly its current rate, so the new value is the old value plus the step times the slope at the start.
Working
The slope at the starting temperature: the bread is cooling at 3.5 °C per minute.
Multiply the rate by the 2-minute step before adding.
Answer
The answer is about 88 degrees Celsius.
Check
The exact solution 25 + 70e−0.05t gives about 88.3 °C at t = 2, just above 88, as expected for cooling that slows down.
Where this goes wrong
Adding the slope without multiplying by the step, 95 − 3.5 = 91.5. The slope is a rate per minute, and the step is 2 minutes.
Leans on
Linear approximation, from Linear approximation
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