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Question

# Find equations for the tangent plane and normal line to the surface given at the prescribed point.$x^{3}+2 x y^{2}-7 x^{3}+3 y+1={0} \text { at } P_{0}(1,1,1)$

Solution

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We have to find the equation of the tangent plane and normal line to the curve

$x^3+2xy^2-7x^3+3y+1=0,$

at $P_0(1,1,1)$. Let $F(x,y)=x^3+2xy^2-7x^3+3y+1$, and consider $C$ to be the level curve $F(x,y)=0$. The gradient $\nabla F$ is normal to $C$ at $P_0$. First we find the gradient of $F$ which is

$\nabla F(x,y)=F_x(x,y)\textbf{i}+F_y(x,y)\textbf{j}.$

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