Chapter 11: Problem 21
Describe the domain and range of the function. $$ f(x, y)=\ln (4-x-y) $$
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In Exercises 47-50, differentiate implicitly to find \(d y / d x\). \(x^{2}-3 x y+y^{2}-2 x+y-5=0\)
In Exercises 35-38, use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results $$ 4 x^{2}-y=6,(2,10) $$
Find a normal vector to the level curve \(f(x, y)=c\) at \(P.\) $$ \begin{array}{l} f(x, y)=6-2 x-3 y \\ c=6, \quad P(0,0) \end{array} $$
Define the gradient of a function of two variables. State the properties of the gradient.
In Exercises \(35-38,\) find \(\partial w / \partial s\) and \(\partial w / \partial t\) using the appropriate Chain Rule, and evaluate each partial derivative at the given values of \(s\) and \(t\) $$ \begin{array}{l} \text { Function } \\ \hline w=x^{2}+y^{2} \\ x=s+t, \quad y=s-t \end{array} $$ $$ \frac{\text { Point }}{s=2, \quad t=-1} $$
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