Introduction to :

Things to remember about Euler's Method:

  • Euler's Method gives only approximate values unless the function happens to be a straight line, in which case Euler's Method gives exact values of the function.
      
  • Euler's method does NOT produce a formula. It generates a numerical solution; that is, approximate values of the function at certain points.

The error in an approximation is the difference between the approximation and the real value of the function.

We can only figure out the error if we know the real value of the function.

Sample Problem

Our first examples of Euler's method used the function y that satisfied the IVP

By thinking backwards we can figure out that

y = x2.

Now we can compute the real value of y at 1:

y(1) = (1)2 = 1.

This lets us figure out how bad our approximations were in some earlier examples.

Using a step size of 0.5 we estimated

y(1) ≅ 0.5.

This has an error of 0.5, since it's 0.5 off from the real answer y(1) = 1.

Using a step size of 0.25 we estimated

y(1) ≅ 0.75.

The error here is 0.25.

Using a step size of 0.2 we estimated

y(1) ≅ 0.8.

This has an error of 0.2.

Practice:

Example 1

Let y = f (x) be a solution to the IVP

Does Euler's method produce an over- or under-estimate for the value of f (2)?


Example 2

Let y = f (x) be a solution to the IVP

For which starting points will Euler's method produce underestimates and for which starting points will Euler's method produce overestimates?


Exercise 1

Let y = f (x) be a solution to the IVP

What is the error when Euler's method is used to estimate f (2) with a step size of 0.5? With a step size of 0.25?


Exercise 2

If f is concave up around x = a, is the tangent line to f at a above or below the graph of f? What about if f is concave down?


Exercise 3

Let y = f (x) be a solution to the IVP

Does Euler's method produce an over- or under-estimate for the value of f (2)?


Exercise 4

Let y = f (x) be a solution to the IVP

Does Euler's method produce an over- or under-estimate for the value of f (3.5)?


Exercise 5

Let y = f (x) be a solution to the IVP

Does Euler's method produce an over- or under-estimate for the value of f (1)?


Exercise 6

Let y = f (x) be a solution to the IVP

Does Euler's method produce an over- or under-estimate for the value of f (2)?


Exercise 7

Let y = f (x) be a solution to the IVP

Does Euler's method produce an over- or under-estimate for the value of f (5.9)?


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