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

What is the equation of the tangent plane to the surface given by the graph of the function \( g(x,y)=1-x^2-y^2\) at the point \((0,0,1)\)?

Find the flux of the vector field \({\bf F}= \bigg\langle \frac{-y}{x^2+y^2},\frac{x}{x^2+y^2},1 \bigg\rangle\) over the cylinder of radius 1, centered around the \(z\)-axis and between \(z=0\) and \(z=1\) with the disks at the top and bottom included so that the surface is closed.

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

Let \(f(x,y,z)=xy+\cos z\) and \({\bf r} (t)=\langle t^2,t,\frac{\pi}{2}\sin^3(t+\frac{\pi}{2}) \rangle\) and let \(g(t)=f({\bf r} (t))\)   Compute \(g'(0).\)

Use Green’s Theorem to compute \(\int_{C} {\bf F} \cdot {\bf dr}\) where \(C\) is the triangle with vertices \((0,2)\), \((-1,-1)\), \((3,-1)\) oriented counter-clockwise, and \({\bf F}= \langle y^{2}+\ln|1+x^{2}|, x(2y+1), e^{\sqrt{1+y^{2}}} \rangle.\)

Let \(C\) be the curve given by the vector function \({\bf r}(t) =\langle 1- \frac{1}{2}t^2, t, t^2\rangle\). Find the equation of the osculating plane to \(C\) at the point \((1,0,0)\).

Compute \(\oint_{C}{\bf F}\cdot d{\bf r}\) where \(C\) is the curve parametrized by \({\bf r}(t)= \langle \cos t,\sin t, \cos t+\sin t \rangle, \quad 0\leq t\leq 2\pi\) and where \({\bf F}=(xy+z){\bf i}+z^2{\bf j}+xyz{\bf k}\) [Hint: The curve lies on the surface \(z=x+y\)]

Evaluate the integral \(\int_{C} {\bf F} \cdot {\bf dr} \) where \({\bf F}=\left( \frac{1}{x}+y \right) {\bf i} + \left( \frac{1}{y} + x \right) {\bf j}\) and \(C\) is a smooth curve with starting point \((1,1)\) and end point \((2,3)\).  

TOTAL QUESTION 6: [10] TOTAL SECTION B: [20] GRAND TOTAL [50]