Calculus II, Spring 2009

Midterm Exam II

May 14, 2009, 10:10 AM. Show all details.

1. Compute

\frac{d}{dy} ∫_{1}^{2+y^{2}} \frac{cos(xy)}{x}dx

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 = 1 \end{matrix}

3. 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}

4. (a) State (need not prove) the second derivative 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 f which has a saddle point at (1, 1).

5. Switch the order of integration of

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

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

6. Express the volume of {ρ ≤ 1} ∩ {ϕ ≤ π/4} as triple integrals in cylindrical and spherical coordinates, respectively. Need not find the numerical values of the integrals.

7. Suppose that f(x, y) is continuous at (0, 0) along each straight line that passes through (0, 0), i.e., if for each a ∈ R,

lim_{t→0} f(t, at) = lim_{t→0} f(at, t) = f(0, 0).

Then does it mean that lim_{(x,y)→(0,0)} f(x, y) = f(0, 0)? Explain.

8. Does the linear approximation of f(x, y) = \sqrt{x^{2} + 3y^{2}} near (0, 0) exist? Explain.

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