91Ó°ÊÓ

Find \(f_{x} f_{y}\) and \(f_{z}\) when \(f(x, y, z)\) is (a) \(x^{2} y+3 y x z-2 z^{3} x^{2} y\) (b) \(\mathrm{e}^{2 z} \cos x y\)

Show that $$ f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} $$ satisfies $$ x f_{x}+y f_{y}+z f_{z}=-f(x, y, z) $$

Show that $$ f(x, y, z)=x+\frac{x-y}{y-z} $$ satisfies $$ f_{x}+f_{y}+f_{z}=1 $$

Find the gradient of \(f(x, y)=x^{2}+2 y^{2}-3 x+2 y\) at the point \((x, y)\) in the direction making an angle \(\alpha\) with the positive \(x\) direction. What is the value of the gradient at \((2,-1)\) when \(\alpha=\frac{1}{6} \pi\) ? What values of \(\alpha\) give the largest gradient at \((2,-1)\) ? The level curve of \(f(x, y)\) through \((2,-1)\) is given by \(f(x, y)=f(2,-1)\). This defines the relationship between \(x\) and \(y\) on the curve. Show that the tangent to the level curve at \((2,-1)\) is perpendicular to the direction of maximum gradient at that point and parallel to the direction of zero gradient.

Find \(\frac{\mathrm{d} A}{\mathrm{~d} t}\) where \(A=r \tan ^{-1}(r \tan \theta)\) and \(r=2 t+1, \theta=\pi t\)

Find \(\partial f / \partial s\) and \(\partial f / \partial t\) when \(f(x, y)=\mathrm{e}^{x} \cos y\) \(x=s^{2}-t^{2}\) and \(y=2 s t\).

Find \(\mathrm{d} z / \mathrm{d} t\) when (a) \(z^{2}=x^{2}+y^{2}, x=t^{2}+1\) and \(y=t-1\) (b) \(z=x^{2} t^{2}\) and \(x^{2}+3 x t+2 t^{2}=1\)

Show that if \(u=x+y, v=x y\) and \(z=f(u, v)\) then (a) \(x \frac{\partial z}{\partial x}-y \frac{\partial z}{\partial y}=(x-y) \frac{\partial z}{\partial u}\) (b) \(\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}=(y-x) \frac{\partial z}{\partial v}\)

Show that if \(z=x^{n} f(u)\), where \(u=y / x\), then $$ x \frac{\partial z}{\partial x}+y \frac{\partial z}{\partial y}=n z $$ Verify this result for \(z=x^{4}+2 y^{4}+3 x y^{3}\)

In a right-angled triangle \(a \mathrm{~cm}\) and \(b \mathrm{~cm}\) are the sides containing the right-angle. \(a\) is increasing at \(2 \mathrm{~cm} \mathrm{~s}^{-1}\) and \(b\) is increasing at \(3 \mathrm{~cm} \mathrm{~s}^{-1}\). Calculate the rate of change of (a) the area and (b) the hypotenuse when \(a=5\) and \(b=3\)