\( y ^ { \prime \prime } + 4 y = 0 \quad y ( 0 ) = 1 \quad y ^ { \prime } ( 0 ) = 1 \)

\[ \text { OR } \] Given a function \( f(x)=\left\{\begin{array}{ll}7-4 x, & x<1 \\ x^{2}+2, & x \geq 1\end{array}\right. \) Evaluate \( \operatorname{limit}_{x \rightarrow-6} f(x) \) and \( \operatorname{limit}_{x \rightarrow 1} f(x) \) if exist.

. Nyuton usulidan foydalanib, tenglamaning ildizini olti xonali kasrgacha to'g'ri hisoblang. . \( f(x)=-x^{3}+4 x^{2}-2 x-14 \rightarrow \min , \varepsilon=0.001 \) xatolik bilan Nyuton usulida optimizatsiya masalasi yechimi topilsin.

Kasr ratsional funksiyalarning integrallarini hisoblang. \[ \int \frac{x+2}{\left(x^{2}-2 x+3\right)^{2}} d x \]

Find the directional derivative of \( \phi x, y, z=x^{2} y z+4 x z^{2} \) at the point \( (1,-2-1) \) in the direction \( 2 i-j-2 k \) Select one: \( \frac{17}{3} \) \( \frac{33}{3} \) \( \frac{37}{3} \) \( \frac{7}{3} \)

Nyuton usulidan foydalanib, tenglamaning ildizini sakkiz xonali kasrgacha to'g'ri hisoblang. Dastlabki taxminlarni topish uchun grafik chizishdan boshlang. f \( f(x)=-x^{3}+4 x^{2}-2 x-14 \rightarrow \min , \varepsilon=0.001 \) xatolik bilan Nyuton usulida optimizatsiya masalasi yechimi topilsin.