Euler's Method

Step forward along the tangent line, over and over.

12 problems · about 23 minutes · F-IF.B.6

NameDate

Figure — use these to answer the problems

The tangent line at the marked point is exactly one Euler step. Notice the curve pulling away from it, which is the error.

y = 1 · 0.75^x + 0

the starting value1

the growth rate0.75

where the step begins1

Figure 1

Straightforward practice. Get the method working first.

5 problems

dy/dx = y with y = 4 and a step of 0.25. What is y after one step?
Answer
2
Where does Euler's error come from?
- The true solution curves away from the tangent line during each step.
- Rounding the arithmetic.
dy/dx = y with y = 2 and a step of 0.5. What is y after one step?
Answer
dy/dx = 4 with y = 1 and a step of 0.25. What is y after one step?
Answer
From x = 0 to x = 3 with a step of 0.5. How many steps?
Answer

The same ideas with more to keep track of.

3 problems

Halving the step size. Does the accumulated error roughly halve? 1 yes, 0 no.
Answer
dy/dx = 3y with y = 1 and a step of 0.2. What is y after one step?
Answer
A solution that is exactly a straight line. Does Euler make any error? 1 yes, 0 no.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Order the steps of one Euler iteration.
Write 1 to 4 in the boxes to put these in order.
- Compute the slope at the current point
- Multiply that slope by the step size
- Add the result to the current y
- Advance x by the step size
dy/dx = 0 with y = 7 and any step size. What is y after one step?
Answer
The First Step
dy/dx = 2y with y = 3 and a step of 0.1. What is y after one step?
Answer
The Step Count
Travelling from x = 0 to x = 2 with a step of 0.25. How many steps are needed?
Answer