Chapter 12.1, Problem 50ES

(a) Sketch the graph of $\text{r}\left(t\right)=〈2t,\frac{2}{1+t^2}〉$

(b) Prove that the curve in part (a) is also the graph of the function $y=\frac{8}{4+x^2}$ [The graphs of $y=a^3/\left(a^2+x^2\right),$ where a denotes a constant, were first studied by the French mathematician Pierre de Fermat, and later by the Italian mathematicians Guido Grandi and Maria Agnesi. Any such curve is now known as a "witch of Agnesi." There are a number of theories for the origin of this name. Some suggest there was a mistranslation by either Grandi or Agnesi of some less colorful Latin name into Italian. Others lay the blame on a translation into English of Agnesi's 1748 treatise, Analytical Institutions.]