Chapter 8.9, Problem 112E

(a)

To determine

To sketch: The graph of the Gaussian function $f\left(x\right)=e^{-a{x}^2}$ for $a=0.5$, $1$ and $2$.

(b)

To determine

To compute: The area under the curve obtained in part (a).

(c)

To determine

To evaluate: The given integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{e}^{-\left(a{x}^2+bx+c\right)}}dx$ by completing the square.