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• Notes

Find an equation of the tangent plane to the given surface at the specified point.
$z=2 x^{2}+y^{2}-5 y,\quad (1,2,-4)$

$z=2 x^{2}+y^{2}-5 y \quad(1,2,-4)$

$f_{x}(x, y)=4 x \quad \quad f_{y}(x, y)=2 y-5$

$f_{x}\left(x_{0}, y_{0}\right)=4 \times 1=4, \quad f_{y}\left(x_{0}, y_{0}\right)=2 \times 2-5=-1$

$z-z_{0}=f_{x}(1,2)[x-1]+f_{y}(1,2)(y-2)$

$z+4=4[x-1]-(y-2) \Rightarrow z=4 x-y-6$

Find the differential of the function.

1- $z=e^{-2 x} \cos 2 \pi t$

2- $m=p^{5} q^{3}$

(1) $z=e^{-2 x} \cos (2 \pi t)$

$d z=\frac{\partial z}{\partial x} d x+\frac{\partial z}{\partial t} d t$

$d z=-2 e^{-2 x} \cos (2 \pi t) d x+2 \pi[-\sin 2 \pi t] e^{-2 x} d t$

(2) $m=p^{5} q^{3}$

$d m=\frac{\partial m}{\partial p} d p+\frac{\partial m}{\partial q} d q=5 p^{4} q^{3} d p+3 q^{2} p^{5} d q$