3 Calculus III

1. Variable $z$ is given in terms of the measurable variables $f$ and $g$ as
   
   $$\frac1z = \frac1f + \frac1g$$
   
   
   The values of $f$ and $g$ and their uncertainties are:
   
   $$f = 4 \pm 0.01 \qquad g = 8 \pm 0.02$$
   
   
   What are the maximum relative and absolute errors in the computed $z$? Are you stunned by value of the relative error in $z$? Explain why not. [1, Total Differential]
2. The position of a particle in two-dimensions is given by
   
   $$x = 2 + 3 t \qquad y = t^2 + 4$$
   
   
   What is the rate at which the distance $r$ from the origin increases at time $t=1$? [1, Total Differential]
3. Find $I_x$ for the area between the curves
   
   $$y = x \qquad y = 4x -x^2$$
   
   
   Exact answers only, please. Since the integrand $y^2$ does not depend on $x$, it would seem logical to integrate $x$ first. Comment on that. [1, Centroids and Moments of Inertia]
4. Find the volume of the region bounded by
   
   $$z=0 \qquad x^2 + y^2 = 4x \qquad x^2 + y^2 = 4 z$$
   
   
   Use cylindrical coordinates $r$, $\theta$ (or $\phi$ if you want), and $z$. What variable is obviously the one to integrate first? For the second integration, discuss each possibility and explain which is the best choice. Would it have been easier to use Cartesian coordinates $x$, $y$ and $z$ instead of cylindrical ones? [1, Triple Integrals]