3 Calculus III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Find the MacLaurin series for $1/(1+x^5)$. Hint: you may not want to crunch this out. Explain why not. Use a suitable trick instead. [1, Taylor and Maclaurin Series]
2. UNGRADED Find $\lim_{x\to0^+} \frac{\ln\cot x}{e^{csc^2 x}}$. [1, L'Hôpital's Rule]
3. UNGRADED Find $\lim_{x\to\pi/2} (\sec^3 x - \tan^3 x)$. [1, L'Hôpital's Rule]
4. Find $\lim_{x\to1} \left(\frac{1}{\ln(x)} - \frac{x}{x-1}\right)$. [1, L'Hôpital's Rule]
5. 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 the value of the relative error in $z$? Explain why not. [1, Total Differential]
6. 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 $r=\sqrt{x^2+y^2}$ from the origin increases at time $t=1$? Do not substitute for $x$ and $y$ inside $r$, leave as is. [1, Total Differential]
7. UNGRADED Use double integration to 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 (since the integral is simply $xy^2$). Discuss that in detail. In particular, draw the region of integration twice. In the first, show the lines of the first integration if you do $x$ first, and in the second show the lines of the first integration if you do $y$ first. Based on those graphs, give complete limits of integration (for both x and y) if you do $x$ first, and the same if you do $y$ first, then compare. Do the double integral that seems easiest to you completely. [1, Centroids and Moments of Inertia]
8. Use double integration in polar coordinates $\rho$ and $\theta$ to find the area outside $\rho=4$ and inside $\rho=8\cos\theta$. Draw the region of integration twice. In the first, show the lines of first integration if you do $\rho$ first (which will be radial line segments as $\theta$ is constamt), and in the second show the lines of first integration if you do $\theta$ first (which will be arcs of circles as $\rho$ is constant). Using that, give complete limits of integration (for both $\rho$ and $\theta$) if you do $\rho$ first, and also if you do $\theta$ first. Then actually do the case that seems easiest to you. Do not integrate over only half of the given region and double it.
9. UNGRADED Use double integration in cartesian coordinates $x$ and $y$ to find the area to the right of $y^2=x$ and below $y=2-x$. In particular, give complete limits of integration if you do $x$ first and if you do $y$ first, then do the one that seems easiest to you.