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Section 14.1 : Tangent Planes and Linear Approximations

1. Find the equation of the tangent plane to \(\displaystyle z = {x^2}{y^4} - \frac{{12x}}{y}\) at \(\left( { - 1,6} \right)\).
2. Find the equation of the tangent plane to \(z = \ln \left( {{x^2}y} \right) - x\,\sqrt y \) at \(\displaystyle \left( { - \frac{1}{2},4} \right)\).
3. Find the equation of the tangent plane to \(z = {{\bf{e}}^{x\,y}} + {y^2}{{\bf{e}}^{1 - y}}\) at \(\left( {0,1} \right)\).
4. Find the linear approximation to \(z = \cos \left( {\sin \left( y \right) - x} \right)\) at \(\left( { - 2,0} \right)\).
5. Find the linear approximation to \(\displaystyle z = \frac{{10{x^2}}}{{x - y}}\) at \(\left( {4, - 1} \right)\).