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Section 12.5 : Functions of Several Variables

For problems 1 – 6 find the domain of the given function.

1. $$f\left( {x,y} \right) = \sqrt {2x + 4y - 1}$$
2. $$\displaystyle f\left( {x,y} \right) = \ln \left( {\frac{1}{{x - y}}} \right)$$
3. $$\displaystyle f\left( {x,y} \right) = \sqrt {\frac{1}{{{x^2}}} - \frac{1}{{{y^2}}}}$$
4. $$\displaystyle f\left( {x,y,z} \right) = \frac{1}{{x + 1}} + \frac{1}{{y - 1}} + \frac{1}{{x + y - z}}$$
5. $$f\left( {x,y,z} \right) = \ln \left( {{x^2} + {y^2} - 8z} \right)$$
6. $$f\left( {x,y} \right) = \sqrt {x + y} - \sqrt {x - 3}$$

For problems 7 – 11 identify and sketch the level curves (or contours) for the given function.

1. $${x^2} - 4z - y = 2$$
2. $$x - 4z - {y^2} = 2$$
3. $${z^2} + 4{x^2} = 1 - 4{y^2}$$
4. $$z + 4{x^2} = 1 - 4{y^2}$$
5. $$2x - 6y + z = - 2$$

For problems 12 – 14 identify and sketch the traces for the given curves.

1. $${x^2} - 4z - y = 2$$
2. $${z^2} + 4{x^2} = 1 - 4{y^2}$$
3. $$2x - 6y + z = - 2$$