\( y ^ { \prime \prime } - y ^ { \prime } - 6 y = 0 \quad y ( 0 ) = 1 \quad y ^ { \prime } ( 0 ) = 2 \)

If \( \phi \) and \( \psi \) are differentiable scalar functions of position \( x, y, z \) then \( \operatorname{grad} \phi+\psi \) is equal to Select one: \( \operatorname{div} \phi+\psi \) \( \operatorname{curl} \phi+\psi \) \( \operatorname{grad} \phi+\operatorname{grad} \psi \) \( \operatorname{div} \phi+\operatorname{div} \psi \)

53) \( \lim _{x \rightarrow \frac{\pi}{3}} \frac{2 \cos x-1}{\sqrt{3}-\tan x} \)

1. Find the fourior series of the fanction \( f(x)=\left\{\begin{array}{ll}-1 ; & -1<x<0 \\ 1 ; & 0<x<1\end{array}\right. \)

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.

2. Evaluate \( \int_{0}^{4 a} \int_{x^{2} / 4 a}^{2 \sqrt{a x}} d y d x \) by changing the order of integration.