Chapter 5.10, Problem 44ES

CAS programs have commands for working with most of the important nonelementary functions. Check your CAS documentation for information about the error function $\text{erf}\left(x\right)$ [see Formula $\left(12\right)$ ], and then complete the following.

(a) Generate the graph of $\text{erf}\left(x\right)$ .

(b) Use the graph to make a conjecture about the existence and location of any relative maxima and minima of $\text{erf}\left(x\right)$ .

(c) Check your conjecture in part (b) using the derivative of $\text{erf}\left(x\right)$ .

(d) Use the graph to make a conjecture about the existence and location of any inflection points of $\text{erf}\left(x\right)$ .

(e) Check your conjecture in part (d) using the second derivative of $\text{erf}\left(x\right)$ .

(f) Use the graph to make a conjecture about the existence of horizontal asymptotes of $\text{erf}\left(x\right)$ .

(g) Check your conjecture in part (f) by using the CAS to find the limits of $\text{erf}\left(x\right)$ as $x\to \pm \infty$ .