Use the Integral Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{1}{k^{2}+1}$

Test the series for convergence or divergence. ∑∞ k=1 k ln k / (k+1)^3

The rate of sales (in sales per month) of a company is given, fort in months since January 1, by $r(t)=t^{4}-20 t^{3}+118 t^{2}-180 t+200$. Graph the rate of sales per month during the first year (t=0 to t=12). Does it appear that more sales were made during the first half of the year, or during the second half?

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19-month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: 1321–1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by 0.14 serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

Compare phase and group velocities for $(\mathrm{a})\ \frac{\partial u}{\partial t}=\beta \frac{\partial^{3} u}{\partial x^{3}}; (\mathrm{b})\ i \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}; (\mathrm{c})\ \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}-u.$ Show that the phase velocity is greater than the sound speed c, but the group velocity is less than the sound speed c.

Multiple Choice. If there is a positive number K for which $\left|s_{n}\right| \leq K$ for all integers $n \geq 1$, then $\left\{s_{n}\right\}$ is [ (a) increasing (b) bounded (c) decreasing (d) convergent ].

A $\mathrm{CO}_{2}$ cartridge is used to propel a small rocket cart. Compressed gas, stored at 35 MPa and $20^{\circ} \mathrm{C}$, is expanded through a smoothly contoured converging nozzle with 0.5 mm throat diameter. The back pressure is atmospheric. Calculate the pressure at the nozzle throat. Evaluate the mass flow rate of carbon dioxide through the nozzle. Determine the thrust available to propel the cart. How much would the thrust increase if a diverging section were added to the nozzle to expand the gas to atmospheric pressure? What is the exit area? Show stagnation states, static states, and the processes on a Ts diagram. Consider the converging-diverging option. To what pressure would the compressed gas need to be raised (keeping the temperature at $20^{\circ} \mathrm{C}$) to develop a thrust of 15N? (Assume isentropic flow.)

Find (a) a symmetric matrix A for the quadratic form; (b) the eigenvalues of A; (c) an orthonormal set of eigenvectors; (d) an orthogonal diagonalizing matrix C. $4 x ;+4 x_{1} x_{2}+x_{2}^{2}$.

Verify that the infinite series diverges. $\sum_{n=1}^{\infty} \frac{(n+1) !}{5 n !}$

Find the indicated limits. $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=-2$, find $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow 0} \frac{f(x)}{x}$.

An incompressible liquid with negligible viscosity and density $\rho=1250 \mathrm{kg} / \mathrm{m}^{3}$ flows steadily through a 5-m-long convergent-divergent section of pipe for which the area varies as $A(x)=A_{0}\left(1+e^{-x / a}-e^{-x / 2 a}\right)$ where $A_{0}=0.25 \mathrm{m}^{2}$ and a=1.5 m. Plot the area for the first 5 m. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first 5 m, if the inlet centerline velocity is 10 m/s and inlet pressure is 300 kPa. Hint: Use relation $u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right)$.

Consider the velocity field $\vec{V}=A\left[x^{4}-6 x^{2} y^{2}+y^{4}\right] \hat{i}+B\left[x^{3} y-x y^{3}\right] \hat{j}$; $A=2 \mathrm{m}^{-3} \cdot \mathrm{s}^{-1}$, B is a constant, and the coordinates are measured in meters. Find B for this to be an incompressible flow. Obtain the equation of the streamline through point (x, y)=(1, 2). Derive an algebraic expression for the acceleration of a fluid particle. Estimate the radius of curvature of the streamline at (x, y)=(1, 2).

Show that the nth derivative of $\left(a x^{2}+b x+c\right) e^{x}$ is a function of the same form but with different constants.

Use partial fractions to find the indefinite integral. $\int \frac{x-8}{x^{2}-x-6} d x$