The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = \(x\) and height = \(3 - x\) for \(0 \leq x \leq 3\). Calculate the volume of the solid when these rectangles are stacked from \(x=0\) to \(x=3\).

Let \( y=e^{x} \ln x \), then \( y^{\prime}= \)

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

Let \( y=\sin x \cdot \cos x \), then \( d y= \)

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 ' \( x \) '.

Let \( y-x e^{y}=1 \), then \( d y(0)= \)