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{\bf Exam Question

Topic: Tangent Plane}

Write down the formula for the directional derivative in the
direction $\theta$ at a point $P$ on the surface given by the
equation $z=f(x,y).$

Prove that the direction of maximum slope and the direction of
zero slope are always at right angles.

Find the equation of the tangent plane at the point $(1,2,-5)$ to
the surface given by the equation $$z=xy^3-3(x+y).$$
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{\bf Solution}

The directional derivative is given by
$$D(\theta)=f_x\cos\theta+f_y\sin\theta,$$ where $f_x$ and $f_y$
are evaluated at $P.$

The direction of zero slope is given by $\tan\theta=-f_x/f_y.$

Now $D'(\theta)=f_x\sin\theta+f_y\cos\theta$ and so $D'(\theta)=0$
when $\tan\theta=f_y/f_x.$

The product of these two directions is $-1$ and so they are at
right-angles.

\begin{eqnarray*}
z&=&xy^2-3(x+y)\\ z_x&=&y^2-3=2^2-3=1\ \ \mathrm{at}\ P\\
z_y&=&2xy-3=2.1.2-3=1\ \ \mathrm{at}\ P
\end{eqnarray*}

The equation of the tangent plane is therefore
$$z+5=1(x-1)+1(y-2);\ \ x+y-z=8.$$

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