Tangent Planes And Linear Approximations

(requires JavaScript)

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

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

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

   $z=-8x-2y$

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

   $z=y\cos <!--FUNCTION\; APPLICATION-->\left(x-y\right)$

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

   $z=y$

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

   $z=y\ln <!--FUNCTION\; APPLICATION-->\left(x\right)$

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

   $f_x<!--FUNCTION\; APPLICATION-->\left(1,4\right)=4$ and $f_y<!--FUNCTION\; APPLICATION-->\left(1,4\right)=0$. 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: $\mathbf{n}={\mathbf{T}}_2\times {\mathbf{T}}_1=⟨4,0,-1⟩$, and so the plane is $4x-z-4=0$.

4. Find the linear approximation of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\sqrt{20-x^2-7y^2}$

   at the point $\left(2,1\right)$ and use it to approximate $f<!--FUNCTION\; APPLICATION-->\left(1.95,1.08\right)$.

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