* stirling's Formula \( P_{n}(x)=y_{0}+\frac{\alpha\left(\Delta y_{0}+\Delta y_{-1}\right)}{2}+\frac{\alpha^{2}}{2!} \Delta^{2} y_{-1}+\frac{\alpha\left(\alpha^{2}-i\right)}{3} \) \( \cdot\left(\frac{\Delta^{3} y_{-1}+\Delta^{3} y_{-2}}{2}\right)+\frac{\alpha^{2}\left(\alpha^{2}-i\right)}{4!} \Delta^{4} y_{-2}+\cdots \) \( \alpha=\frac{x-x_{0}}{h}, h=x_{i+1}-x_{i} \) \( P_{n}(x)=\frac{1}{h}\left[\left(\frac{\Delta y_{0}+\Delta y_{-1}}{2}\right)-\frac{1}{6}\left(\frac{\Delta^{3} y_{-1}+\Delta^{3} y_{-2}}{2}\right)\right. \) \( +\frac{1}{30}\left[\frac{\Delta^{5} y_{-2}+\Delta^{5} y_{-3}}{2}\right]+\cdots \)

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

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.

If a fo \( f(x) \) sefinet beloe is contincious at \( x=0 \) Find the volue of \( k \). \( f(x)=\left\{\begin{array}{l}\frac{1-\cos 2 x}{2 x^{2}}, x<0 \\ k, x=0 \\ \frac{x}{|x|}, x>0 .\end{array}\right. \)

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

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