Accuracy and Usefulness of Euler's Method Examples

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)?

We left out the number of steps to use with Euler's method, but that doesn't matter for the answer. The slope field for looks like this:

Since the initial condition puts the solution under the line y = x, the solution is concave down:

When a function is concave down, all the tangent lines will lie above the graph.

Therefore Euler's method will produce an overestimate.

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?

The slope field looks like this:

A non-zero solution to this d.e. must lie either entirely above or entirely below the x-axis. If a solution lies above, it is concave up. If a solution lies below, it is concave down.

For any starting point of the form (x, y) where y > 0, Euler's method will produce an underestimate.

For any starting point of the form (x, y) where y < 0, Euler's method will produce an overestimate.