Use the Limit Comparison Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{k+5}{k^{k+1}}$

A pitot static probe is placed in a converging-diverging duct through which air flows. The duct is fed by a reservoir kept at $20^{\circ} \mathrm{C}$. If the probe reads a static pressure of 75 kPa and a stagnation pressure of 100 kPa at a location where the area is 0.00645 $\mathrm{m}^{2}$, what is the local velocity and the mass flow rate of air?

A newspaper headline stated that at a recent budget workshop, nearly three dozen people supported a sales tax increase to help deal with the city’s financial deficit (San Luis Obispo Tribune, January 22, 2005). This conclusion was based on data from a survey acknowledged to be unscientific, in which 34 out of the 43 people who chose to attend the budget workshop recommended raising the sales tax. Briefly discuss why the survey was described as “unscientific” and how this might limit the conclusions that can be drawn from the survey data.

Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

$$\sum _ { n = 2 } ^ { \infty } \frac { 1 } { 5 n + 10 \sqrt { n } }$$

Use the Integral Test to determine if the series converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

$$\sum _ { n = 1 } ^ { \infty } e ^ { - 2 n }$$

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations. $P(0,0,0), Q(5,2,2)$

In each case, find a linear differential equation of second order satisfied by $u_1$ and $u_2.$ (a) $u_{1}(x)=e^{x}, \ u_{2}(x)=e^{-x}$. (b) $u_{1}(x)=e^{2 x}, \ u_{2}(x)=x e^{2 x}$. (c) $u_{1}(x)=e^{-x / 2} \cos x, \ u_{2}(x)=e^{-x / 2} \sin x$. (d) $u_{1}(x)=\sin (2 \mathrm{x}+1), \ u_{2}(x)=\sin (2 \mathrm{x}+2)$. (e) $u_{1}(x)=\cosh \mathrm{x}, \ u_{2}(x)=\sinh \mathrm{x}$.

Consider a flow field represented by the stream function $\psi=-\mathrm{A} / 2\left(x^{2}+y^{2}\right)$, where A=constant. Is this a possible twodimensional incompressible flow? Is the flow irrotational?

Convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. $x^{2}+y^{2}=45$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Every point in an open region is an interior point.

Find the curvature and radius of curvature of the plane curve at the given value of x. $y=x-\frac{4}{x}$, x=2

Simplify the rational expression. $\frac{x^{3}+9 x^{2}+2 x-48}{x-2}$

Let V be the set of all polynomial $p(t)=a_{0}+a_{1} t+a_{2} t^{2}$ which satisfy $p(0)+2 p^{\prime}(0)+3 p^{\prime \prime}(0)=0.$ Show that V is a vector space and find a basis for it.

Determine whether each geometric series converges or diverges. If it converges, find its sum. $1+\frac{1}{3}+\frac{1}{9}+\dots+\left(\frac{1}{3}\right)^{n}+\cdots$