A region is defined by the functions \( y = e^{-x} \) for \( x \geq 0 \) and \( y = 0 \). What is the volume generated when this region is revolved around the line \( y = 1 \)?

If \( x^{3}+y^{3}=6 x y \), then find \( \frac{d y}{d x} \) and \( \frac{d^{2} y}{d x^{2}} \)

Ex. 1. A company manufactures and sales \( X \) units of radio transistor per week. If the weekly cost and demand equations are: \[ C(X)=5,000+2 x \] \[ P(X)=10-\frac{x}{1,000} \quad P \text { is price } \quad[0,8000] \] or each week find:- 1) The production level that leads to maximum revenue and the maximum revenue. 2) The production levels that leads to maximum profit and the maximum profit. 3) The production level that leads to minimum cost and the minimum cost.

22. a) If \( f(x)=\frac{|6-x|}{x-6} \), find \( \lim _{x \rightarrow 6+} f(x) \) (if exist.) b) Evaluate: \( \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}} \) c) Evaluate: \( \lim _{x \rightarrow 0} \frac{x\left(2^{2}-1\right)}{1-\cos x} \) KMC Note d) Find \( \frac{d x}{d y} \) if \( x^{3}+y^{3}=3 a x^{2} y \) e) What is the derivative of \( \sec ^{-1} x \) with respect to ' \( x \) '.

4. (15б) Выясните, является ли функция \( z=\ln \left(x^{2}+y^{2}+2 y+1\right) \) решением уравнения \( \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 \)

\( \left. \begin{array} { l }{ \frac { d } { d x } ( x ^ { 2 } \sin ^ { 2 } x ^ { 2 } ) } \\ { = x ^ { 2 } \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) + \sin ^ { 2 } x ^ { 2 } \frac { d } { d x } ( x ^ { 2 } ) } \\ { = x ^ { 2 } \frac { d } { d x } ( \sin x ) ^ { 2 } + \sin ^ { 2 } x ^ { 2 } \frac { d } { d x } ( x ^ { 2 } ) } \\ { = x ^ { 2 } ( 2 \cos x ) + \sin ^ { 2 } x ^ { 2 } 2 x } \\ { = 2 x ^ { 2 } \cos x + 2 x \sin ^ { 2 } x ^ { 2 } } \end{array} \right. \)