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7.3. EULER’S METHOD
Summary
In this section, we encountered the following important ideas:
• Euler’s method is an algorithm for approximating the solution to an initial value
problem by following the tangent lines while we take horizontal steps across the t-axis.
• If we wish to approximate y(t) for some fixed t by taking horizontal steps of size ∆t,
then the error in our approximation is proportional to ∆t.
Exercises
1. Newton’s Law of Cooling says that the rate at which an object, such as a cup of coffee,
cools is proportional to the difference in the object’s temperature and room temperature.
If T(t) is the object’s temperature and T r is room temperature, this law is expressed at
dT
dt
= −k(T − T r ),
where k is a constant of proportionality. In this problem, temperature is measured in
degrees Fahrenheit and time in minutes.
(a) Two calculus students, Alice and Bob, enter a 70 ◦ classroom at the same time.
Each has a cup of coffee that is 100 ◦ . The differential equation for Alice has a
constant of proportionality k = 0.5, while the constant of proportionality for
Bob is k = 0.1.
What is the initial rate of change for Alice’s coffee? What is the initial rate of
change for Bob’s coffee?
(b) What feature of Alice’s and Bob’s cups of coffee could explain this difference?
(c) As the heating unit turns on and off in the room, the temperature in the room
is
T r = 70 + 10 sin t.
Implement Euler’s method with a step size of ∆t = 0.1 to approximate the
temperature of Alice’s coffee over the time interval 0 ≤ t ≤ 50. This will most
easily be performed using a spreadsheet such as Excel. Graph the temperature
of her coffee and room temperature over this interval.
(d) In the same way, implement Euler’s method to approximate the temperature of
Bob’s coffee over the same time interval. Graph the temperature of his coffee
and room temperature over the interval.
(e) Explain the similarities and differences that you see in the behavior of Alice’s
and Bob’s cups of coffee.
7.3. EULER’S METHOD
Summary
In this section, we encountered the following important ideas:
• Euler’s method is an algorithm for approximating the solution to an initial value
problem by following the tangent lines while we take horizontal steps across the t-axis.
• If we wish to approximate y(t) for some fixed t by taking horizontal steps of size ∆t,
then the error in our approximation is proportional to ∆t.
Exercises
1. Newton’s Law of Cooling says that the rate at which an object, such as a cup of coffee,
cools is proportional to the difference in the object’s temperature and room temperature.
If T(t) is the object’s temperature and T r is room temperature, this law is expressed at
dT
dt
= −k(T − T r ),
where k is a constant of proportionality. In this problem, temperature is measured in
degrees Fahrenheit and time in minutes.
(a) Two calculus students, Alice and Bob, enter a 70 ◦ classroom at the same time.
Each has a cup of coffee that is 100 ◦ . The differential equation for Alice has a
constant of proportionality k = 0.5, while the constant of proportionality for
Bob is k = 0.1.
What is the initial rate of change for Alice’s coffee? What is the initial rate of
change for Bob’s coffee?
(b) What feature of Alice’s and Bob’s cups of coffee could explain this difference?
(c) As the heating unit turns on and off in the room, the temperature in the room
is
T r = 70 + 10 sin t.
Implement Euler’s method with a step size of ∆t = 0.1 to approximate the
temperature of Alice’s coffee over the time interval 0 ≤ t ≤ 50. This will most
easily be performed using a spreadsheet such as Excel. Graph the temperature
of her coffee and room temperature over this interval.
(d) In the same way, implement Euler’s method to approximate the temperature of
Bob’s coffee over the same time interval. Graph the temperature of his coffee
and room temperature over the interval.
(e) Explain the similarities and differences that you see in the behavior of Alice’s
and Bob’s cups of coffee.