Calculus II, Spring 2014

Quiz 4

May 01, 2014

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1. Let f(x, y) = x^{2} + xy + y^{2}. Find the direction (cos θ, sin θ) for which the directional derivative D_{(cos θ,sin θ)}f(1, 1) is smallest.

2. 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}

3. Find all critical points of f(x, y) = x^{3}y^{2} and determine whether they are local min, local max or neither.

4. Use Lagrangian multipliers (and not other methods) to find extreme values of f(x, y, z) = xy + z^{2} on x^{2} + y^{2} + z^{2} = 16.

5. Suppose that f(x, y) and all its partial derivatives of any order are all continuous. \underline{DERIVE} Taylor’s formula at (0, 0) up to second order. That is, f(x, y) = p_{2}(x, y) + R_{2}(x, y), derive formula for the quadratic polynomial p_{2} and the remainder term R_{2}.

Calculus II, Spring 2014

Quiz 4

May 01, 2014

Show all details.

1. Let f(x, y) = x^{2} + xy + y^{2}. Find the direction (cos θ, sin θ) for which the directional derivative D_{(cos θ,sin θ)}f(1, 1) is smallest.

2. 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}

3. Find all critical points of f(x, y) = x^{3}y^{2} and determine whether they are local min, local max or neither.

4. Use Lagrangian multipliers (and not other methods) to find extreme values of f(x, y, z) = xy + z^{2} on x^{2} + y^{2} + z^{2} = 16.

5. Suppose that f(x, y) and all its partial derivatives of any order are all continuous. \underline{DERIVE} Taylor’s formula at (0, 0) up to second order. That is, f(x, y) = p_{2}(x, y) + R_{2}(x, y), derive formula for the quadratic polynomial p_{2} and the remainder term R_{2}.