Tangent Planes And Linear Approximations

(requires JavaScript)

1. Find an equation of the plane tangent to the surface

$z=4{x}^{2}-{y}^{2}+2y$

at the point $\left(-1,2,4\right)$.

$z=-8x-2y$
2. Find an equation of the plane tangent to the surface

$z=y\mathrm{cos}\left(x-y\right)$

at the point $\left(2,2,2\right)$.

$z=y$
3. Find an equation of the plane tangent to the surface

$z=y\mathrm{ln}\left(x\right)$

at the point $\left(1,4,0\right)$.

${f}_{x}\left(1,4\right)=4$ and ${f}_{y}\left(1,4\right)=0$. Construct $2$ vectors tangent to the surface: ${\mathbf{T}}_{1}=⟨1,0,4⟩$ and ${\mathbf{T}}_{2}=⟨0,1,0⟩$. Find a vector orthogonal to the surface: $\mathbf{n}={\mathbf{T}}_{2}×{\mathbf{T}}_{1}=⟨4,0,-1⟩$, and so the plane is $4x-z-4=0$.

4. Find the linear approximation of the function

$f\left(x,y\right)=\sqrt{20-{x}^{2}-7{y}^{2}}$

at the point $\left(2,1\right)$ and use it to approximate $f\left(1.95,1.08\right)$.

$L\left(x,y\right)=-\frac{2}{3}x-\frac{7}{3}y+\frac{20}{3}$,
$L\left(1.95,1.08\right)=\frac{427}{150}$.