13.7-bo'lim: Yo'nalishli hosilalar

1 va 2 masalalar uchun berilgan funksiyaning gradientini aniqlang.

1. \(\displaystyle f\left( {x,y} \right) = {x^2}\sec \left( {3x} \right) - \frac{{{x^2}}}{{{y^3}}}\) Yechim
2. \(f\left( {x,y,z} \right) = x\cos \left( {xy} \right) + {z^2}{y^4} - 7xz\) Yechim

3 va 4-masalalar uchun ko'rsatilgan yo'nalishda berilgan funktsiya uchun \({D_{\vec u}}f\) ni aniqlang.

1. \(\displaystyle f\left( {x,y} \right) = \cos \left( {\frac{x}{y}} \right)\) \(\vec v = \left\langle {3, - 4} \right\rangle \) yo'nalishida Yechim
2. \(f\left( {x,y,z} \right) = {x^2}{y^3} - 4xz\) \(\vec v = \left\langle { - 1,2,0} \right\rangle \) yo'nalishida Yechim
3. \(f\left( {x,y,z} \right) = 4x - {y^2}{{\bf{e}}^{3x\,z}}\) uchun \({D_{\vec u}}f\left( {3, - 1,0} \right)\) ni \(\vec v = \left\langle { - 1,4,2} \right\rangle \) yo‘nalishida aniqlang. Yechim

6 va 7-masalalar uchun funktsiyaning ko'rsatilgan nuqtadagi maksimal o'zgarish tezligini va bu maksimal o'zgarish tezligi sodir bo'lgan yo'nalishni toping.

1. \(f\left( {x,y} \right) = \sqrt {{x^2} + {y^4}} \) at \(\left( { - 2,3} \right)\) Yechim
2. \(f\left( {x,y,z} \right) = {{\bf{e}}^{2x}}\cos \left( {y - 2z} \right)\) at \(\left( {4, - 2,0} \right)\) Yechim