Show All Notes Hide All Notes

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. \(2x - 6y + z = - 2\)