Calculus II, Spring 2013

Midterm Exam 2

May 09, 2013, 10:10 AM. Show all details.

1. (20 pts) Let f(x, y) = \frac{x^{2}y}{x^{2} + y^{2}} , for (x, y) ≠ (0, 0) and f(0, 0) = 0. P = (0, 0) and

u^{θ} = (cos θ, sin θ), θ ∈ [0, 2π].

(a) For fixed θ, write down the definition of the directional derivative (\frac{df}{ds})_{u^{θ},P} and

evaluate it.

(b) Does f have a linear approximation near (0, 0)? Explain.

2. (12 pts) Use Lagrangian multipliers (and only Lagrangian multipliers) to find extreme values of f(x, y, z) = xy + 2z^{2} on

\{ \begin{matrix}x^{2} + y^{2} + z^{2} = 9 \\ x − y = 0 \end{matrix}

3. (16 pts)

(a) State (need not prove) the second derivative test (discriminant test) for f(x, y). You may assume that f and all its first and second derivatives are continuous in R^{2}.

(b) Give an example of a function w = g(x, y, z) which has a saddle point at (x, y, z) = (0, 0, 0).

4. (12 pts) Evaluate

∫_{0}^{2} ∫_{y}^{2} x^{2} cos(xy) dxdy

5. (12 pts) Switch the order of integration of

∫_{−1}^{1} ∫_{x^{2}}^{1} ∫_{0}^{1−y} dzdydx

to dydxdz and dxdydz respectively. You don’t need to find the numerical value of the integral.

6. (20 pts) Express the volume of {ρ ≤ 1} ∩ {ϕ ≤ π/4} as triple integrals in cylindrical and spherical coordinates, respectively. Then find the volume.

7. (12 pts) Find the Taylor expansion of f(x, y, z) around (x_{0}, y_{0}, z_{0}) up to quadratic terms of x, y and z. Give an expression of the remainder term, R_{2} = \frac{1}{3!}(⋯). You may assume that f and all its first and second derivatives are continuous in R^{3}.