Calculus, Spring 2009

Quiz 4

May 07, 2009 Show all details. 1. Find all local maxima, local minina and saddle points of f(x, y) = x^{4} + y^{4} + 4xy.

2. Use Lagrangian multipliers (and only Lagrangian multipliers) to find extreme values

of f(x, y, z) = xy + z^{2} on

\{ \begin{matrix}x^{2} + y^{2} + z^{2} = 16 \\ x − y = 0 \end{matrix}

3. Find the equation of plane normal to the following curve at (1, 1, −1)

\{ \begin{matrix}x^{2} + 2y^{2} + 3z^{2} = 6 \\ x + y − z = 3 \end{matrix}

4. Evaluate

∫_{0}^{2} ∫_{x}^{2} y^{2} sin xydydx

5. Evaluate

∫_{−1}^{1} ∫_{−\sqrt{1−x^{2}}}^{√\overline{1−x^{2}}} \frac{2dydx}{(1 + x^{2} + y^{2})^{2}}

May 07, 2009 Show all details. 1. Find all local maxima, local minina and saddle points of f(x, y) = x^{4} + y^{4} + 4xy.

2. Use Lagrangian multipliers (and only Lagrangian multipliers) to find extreme values

of f(x, y, z) = xy + z^{2} on

\{ \begin{matrix}x^{2} + y^{2} + z^{2} = 16 \\ x − y = 0 \end{matrix}

3. Find the equation of plane normal to the following curve at (1, 1, −1)

\{ \begin{matrix}x^{2} + 2y^{2} + 3z^{2} = 6 \\ x + y − z = 3 \end{matrix}

4. Evaluate

∫_{0}^{2} ∫_{x}^{2} y^{2} sin xydydx

5. Evaluate

∫_{−1}^{1} ∫_{−\sqrt{1−x^{2}}}^{√\overline{1−x^{2}}} \frac{2dydx}{(1 + x^{2} + y^{2})^{2}}