Chapter 1.1, Problem 59E

Consider the differential equation dy/dx = $e^{-x^2}$.

1. (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis.
2. (b) What are $\underset{x\to -\infty }{\lim }\text{ d}y\text{/d}x$ and $\underset{x\to \infty }{\lim }\text{ d}y\text{/d}x$? What does this suggest about a solution curve as $x\to \pm \infty$
3. (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up.
4. (d) Sketch the graph of a solution y = ϕ(x) of the differential equation whose shape is suggested by parts (a) − (c).