List of practice Questions

The value of the integral \(I = \int_0^1\int_0^1 x\cos(xy)\,dx\,dy\) is

A unit vector that maximizes the directional derivative of the function \(f(x,y)=g(2x+y)\), where \(g'(3)=3\), at the point (1,1) is ____.

What is the value of the integral \(\iint_D dA\), where D is the region bounded by the curve \(y=x^2\) and the line \(y=4\)?

The iterated integral \[\int_0^2\int_0^{4-y^2}\int_0^{2-x} dz\,dy\,dx\] represents the volume of which one of the following solid regions?

For the function \(f(x,y)=x^3+y^3-3xy\), consider the following statements:
(I) f has 3 critical points.
(II) f has two saddle points and one point for local minimum.
(III) f has only one saddle point, a point for local minimum and a point for local maximum.
Choose the correct answer.

Consider the function \[f(x,y)=\begin{cases}(x^2+y^2)\sin\dfrac{1}{x^2+y^2}, & (x,y)\neq(0,0)\\ 0, & (x,y)=(0,0)\end{cases}\] Which one of the following statements is correct for this function?