Examen de Matemàtiques URV 2021 · Segon Parcial 2021

Consider the two-variable function:

$$f(x, y) = \frac{y^3}{3} - \frac{xy^2}{2} + \frac{2x^3}{9} - \frac{x}{6}$$

a)

Compute all first-order and second-order partial derivatives of $f(x, y)$.

b)

Find all the critical points of $f$ and classify each of them as a local maximum, local minimum, or saddle point using the Hessian matrix.

Critical point $(x, y)$

Classification

$(-1/2, 0)$

$(1/2, 0)$

$(-1, -1)$

$(1, 1)$

Problem 2. Double Integral over a Region Bounded by Conics

Consider the region $D \subset \mathbb{R}^2$ defined by:$$D = \left\{ (x,y) \in \mathbb{R}^2: \frac{x^2}{9} + \frac{y^2}{4} \le 1, \quad \frac{x^2}{4} - y^2 \ge 1, \quad x \le 0, \quad y \le 0 \right\}$$

a)

Identify the geometric curves bounding the domain $D$ and find their intersection points in the third quadrant.

$x =$ $y =$

b)

Set up the double integral $\iint_D xy\,dA$ by integrating with respect to $x$ first (horizontal slices) and by integrating with respect to $y$ first (vertical slices).

Problem 3. Second-Order Linear Differential Equations

a.

Find the general solution of the homogeneous linear differential equation:

$$y'' - 3y' - 4y = 0$$

b.

Find the general solution of the non-homogeneous equation:

$$y'' - 3y' - 4y = -3\cos(2t) + \sin(2t)$$