Tangent Planes and Linear Approximations

  1. Find an equation of the tangent plane to the surface defined by the curve \( xy+yz+zx=11\) at the point \((1,2,3)\).    solution

  2. Find an equation of the tangent plane to the curve \(z=\ln(2x+y) \) at the point \((1,-1) \).   solution

  3. Find the linear approximation of the curve \( z=\tan^{-1}(x+2y)\) at the point \((1, 0)\).   solution

  4. Find the linear appproximation of the curve \(f(x,y)=\sqrt{26-x^2-4y^2} \) at the point \((1, 2)\), and use it to approximate the function at \((1.08, 2.05)\).  solution