(a) Sketch the graph ofr(t) =2t,21 + t 2(b) Prove that the curve in part (a) is also the

Chapter 12, Problem 50

(choose chapter or problem)

(a) Sketch the graph of

                                           \(r(t)=\left\langle 2 t, \frac{2}{1+t^{2}}\right\rangle\)

(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.]

Equation Transcription:

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Text Transcription:

r(t)=left angle 2t,2/1+t^2right angle

y=8/4 + x^2

y=a^3/(a^2+x^2)