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### Section 2-3 : Interpretations of Partial Derivatives

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