An interesting biological experiment (see [Schr2]) concerns the determination ofthe maximum water temperature, Xm, at which various species of hydra can survive without shortened life expectancy. One approach to the solution ofthis problem uses a weighted least squares fit ofthe form f(x) = y = a/{x b)c to a collection of experimental data. The x-values ofthe data refer to water temperature. The constant b is the asymptote ofthe graph of / and as such is an approximation to Xm2 a. Show that choosing a, b, and c to minimize Y^=i nonlinear system w/.v,- Ui-by- reduces to solving the w'/V, ^(xi-hy/ ^(xi-h)*' E 1 = E i=\ n o-Y. W/V, {Xi-bYr-E^ (xi - h)2l+i E wiy-, VV,y; (=1 n (Xi - b)7-E ln(x/ - h) U,- - A)24" /=1 U' - ^)t+l /=1 w, y; ln(x, - b) E I Ew,y,- i ((x,- r- by E ix, - by*' i (X,- - A)2 '- /=l v ' ' /=! ' i=l v ' 7 (=1 b. Solve the nonlinear system for the species with the following data. Use the weights w,- = In y,-. i 1 2 3 4 >'/ 2.40 3.80 4.75 21.60 Xi 31.8 31.5 31.2 30.2

Page 1 of 3 Panel 2 Panel 4 10/20/2015 : OcPanel 1NEAR ALGEBRA (Session) Panel 3 yw2jw Page 2 of 3 Panel 6 Panel 8 10/20/2015 : OcPanel 5NEAR ALGEBRA (Session) Panel 7 yw2jw