13.7-bo'lim : Yo'nalishli hosilalar
6. \(f\left( {x,y} \right) = \sqrt {{x^2} + {y^4}} \) ning \(\left( { - 2,3} \right)\) da maksimal o‘zgarish tezligini va bu maksimal o‘zgarish tezligi qaysi yo‘nalishda sodir bo‘lishini toping.
Yechim
Birinchidan, bizga \(\left( { - 2,3} \right)\) da gradient va uning qiymati kerak bo'ladi.
\[\nabla f = \left\langle {\frac{x}{{\sqrt {{x^2} + {y^4}} }},\frac{{2{y^3}}}{{\sqrt {{x^2} + {y^4}} }}} \right\rangle \hspace{0.5in}\nabla f\left( { - 2,3} \right) = \left\langle {\frac{{ - 2}}{{\sqrt {85} }},\frac{{54}}{{\sqrt {85} }}} \right\rangle \]2-qadam
Endi, sinfdagi teorema bo'yicha biz bilamizki, ko'rib chiqilayotgan nuqtadagi maksimal o'zgarish tezligi oddiygina \(\left( { - 2,3} \right)\) gradienti bo'lib, biz oldingi bosqichda topdik. Demak, funktsiyaning maksimal o'zgarish tezligi sodir bo'ladigan yo'nalish,
\[\nabla f\left( { - 2,3} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\left\langle {\frac{{ - 2}}{{\sqrt {85} }},\frac{{54}}{{\sqrt {85} }}} \right\rangle }}\]3-qadam
O'zgarishning maksimal tezligi oddiygina oldingi bosqichdagi gradientning kattaligidir. Shunday qilib, funktsiyaning maksimal o'zgarish tezligi,
\[\left\| {\nabla f\left( { - 2,3} \right)} \right\| = \sqrt {\frac{4}{{85}} + \frac{{2916}}{{85}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\sqrt {\frac{{584}}{{17}}} = 5.8611}}\]