Calculus II, Fall 2013

Quiz 4

May 02, 2013

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1. Let z = L(x, y) be a plane. Write down (need not prove) the definition of ’L(x, y) is a tangent plane of z = f(x, y) at (0, 0, f(0, 0))’.

2. 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}.

3. 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.

4. 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}

5. Evaluate

∫_{0}^{2} ∫_{x}^{2} y^{2} sin(xy) dydx

Calculus II, Fall 2013

Quiz 4

May 02, 2013

Show all details.

1. Let z = L(x, y) be a plane. Write down (need not prove) the definition of ’L(x, y) is a tangent plane of z = f(x, y) at (0, 0, f(0, 0))’.

2. 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}.

3. 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.

4. 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}

5. Evaluate

∫_{0}^{2} ∫_{x}^{2} y^{2} sin(xy) dydx