# Differential Equations

# Accuracy and Usefulness of Euler's Method Exercises

### Example 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?

### Example 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?

### Example 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)?

### Example 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)?

### Example 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)?

### Example 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)?

### Example 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)?