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?

Answer

If *f* is concave up, the tangent line to *f* at a is below the graph of *f*. This is true whether the slope of the tangent line is negative, zero, or positive.

The moral of the previous exercise is that when *f* is concave up, a tangent line approximation is an underestimate. Since Euler's method is a bunch of tangent line approximations stuck together, Euler's method will also provide an underestimate, regardless of how many steps are used

When *f* is concave down, a tangent line approximation is an overestimate. Since Euler's method is a sequence of tangent line approximations, Euler's method also provides an overestimate, regardless of how many steps are used.

Remember when we drew, on a slope field, solutions to a differential equation that passed through a particular point? If we can sketch a solution, we can tell whether it's concave up or concave down. That's all we need to tell whether Euler's method will overestimate or underestimate.