Examen de Matemàtiques URV 2025 · Exemples

Question 1 (4 pts). Domain and local extrema of a two-variable function

Consider the function of two variables defined by:

$$f(x, y) = x\sqrt{1 + 3y} + y\sqrt{1 + 3x}$$

a.

Determine the domain of $f$ and provide a sketch of it in the $xy$-plane.

b.

Locate all the local extrema of $f$, if any exist.

(,)

Question 2 (3 pts). Volume under a surface using double integration

Calculate the volume under the surface given by the graph of $f(x, y) = 4y$ over the bounded region of the plane enclosed between the line $y = x$ and the parabola $x = y^2 - 2$.

a)

Limits of integration with respect to $y$:

b)

c)

Question 3 (3 pts). Vertical motion with aerodynamic drag

An object of mass $m$ travels vertically through the atmosphere. The forces acting on the body are solely its weight and aerodynamic air resistance, which is directly proportional to its velocity (drag constant $k$). Consequently, the differential equation governing the velocity as a function of time is:$$m\frac{dv}{dt} = -mg - kv$$where $g = 9.8\text{ m/s}^2$. Prior observations establish that in the long run the object tends to fall at a constant terminal speed of $10\text{ m/s}$.

a)

Find the numerical value of the ratio $m/k$ for this object.

b)

Assuming the object is launched vertically upwards with an initial velocity $v(0) = v_0$, find an expression in terms of $v_0$ for the time during which the object continues to ascend.

c)

Assuming the initial height at $t = 0$ is zero ($x(0) = 0$), determine the height $x(t)$ as a function of time and the parameter $v_0$, and compute the maximum height attained by the object.