Blog

Find the absolute maximum of the function \(f(x,y,z)=x^2-2y^2+z^2\) on the ellipsoid \(\frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{25}=1\)

Compute the following integral \(\iint_{D} e^{-x^2-y^2}dA\), where \(D\) is the set \(D=\{ (x,y) |\quad 1\leq x^2+y^2 \leq 4, \quad x\geq 0,\quad y\geq 0 \}\)

Compute the directional derivative of \(f(x,y)=x^2+y^2\) at the point \(p=(2,1)\) in the direction of the vector \({\bf v}= \langle 1,1 \rangle\)

Let \(E\) be a solid with volume equal to \(2.\) Let \(S\) be it’s boundary. Compute: \(\iint_{S} {\bf F}\cdot {\bf n} dS\) where \({\bf F}=y{\bf i}+x{\bf j}+3z{\bf k}\) and \({\bf n}\) is oriented outwards.

Calculate \(\frac{\partial(x,y)}{\partial(u,v)}\) if \(u=2x+y\) and \(v=x+\ln(2y)\).

Compute the flux of the vector field \({\bf F}=yz^2{\bf i}+x^2\cos z{\bf j}+xy{\bf k}\) over the paraboloid parametrized by \({\bf r}(u,v)=\langle u,v,1-u^2-v^2 \rangle,\quad u^2+v^2\leq 1\) with downward orientation. [HINT: Try integrating over a different surface]

Which of the following is the form of the particular solution of the differential equation \(y”-4y’+4y=e^{2t}\cos(t)\)as given by the method of undetermined coefficients?

Consider the triangle with vertices \(P=(1,0,1)\), \(Q=(3,0,0)\), and \(R=(2,-1,4)\). (i) Find a vector \(\mathbf{v}\) orthogonal to the triangle \(PQR\) (ii) Find the area of \(PQR\).

Find the domain of the vector function \({\bf r}(t)=\langle \ln(4-t^{2}), \sqrt{t+1} \rangle\)

Determine the length of the curve \({\bf r}(t) = \langle 2t, 3sin(2t), 3cos(2t) \rangle\) on the interval \(0 \leq t \leq \pi\).