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.
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.
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