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Section 13.3 : Interpretations of Partial Derivatives

1. Determine if \(f\left( {x,y} \right) = 10 - {x^2} - {y^2}\) is increasing or decreasing at \(\left( {7, - 3} \right)\) if
   1. we allow \(x\) to vary and hold \(y\) fixed.
   2. we allow \(y\) to vary and hold \(x\) fixed.
2. Determine if \(f\left( {x,y} \right) = x{{\bf{e}}^{x - y}} + 100y\) is increasing or decreasing at \(\left( { - 2,1} \right)\) if
   1. we allow \(x\) to vary and hold \(y\) fixed.
   2. we allow \(y\) to vary and hold \(x\) fixed.
3. Determine if \(\displaystyle f\left( {x,y} \right) = \frac{{x + y}}{{y - x}}\) is increasing or decreasing at \(\left( {0,7} \right)\) if
   1. we allow \(x\) to vary and hold \(y\) fixed.
   2. we allow \(y\) to vary and hold \(x\) fixed.
4. Write down the vector equations of the tangent lines to the traces for \(f\left( {x,y} \right) = \sin \left( x \right)\cos \left( y \right)\) at \(\displaystyle \left( {\frac{\pi }{3}, - \frac{\pi }{4}} \right)\).
5. Write down the vector equations of the tangent lines to the traces for \(\displaystyle f\left( {x,y} \right) = \ln \left( {\frac{x}{{{y^2}}}} \right)\) at \(\left( {6,2} \right)\).