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 $u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right)$.

The following table gives the total world emissions of $\mathrm{CO}_{2}$ from fossil fuels, in billions of tons per year.

$$\begin{array}{c|c|c|c|c|c|c|} \hline {\text { Year }} & {1981} & {1986} & {1991} & {1996} & {2001} & {2006} & {2011} \\ \hline \mathrm{CO}_{2(\mathrm{bn}\text { tons per year) } } & {20.1} & {21.3} & {21.9} & {23.4} & {24.2} & {29.1} & {34.7} \\ \hline \end{array}$$

Use this data to estimate the total world $CO_2$ emissions between 1981 and 2011 using a right sum with n=3.

An insulation company is examining a new material for extruding into cavities. The experimental data is given below for the speed U of the upper plate, which is separated from a fixed lower plate by a 1-mm-thick sample of the material, when a given shear stress is applied. Determine the type of material. If a replacement material with a minimum yield stress of 250 Pa is needed, what viscosity will the material need to have the same behavior as the current material at a shear stress of 450 Pa?

$$\begin{matrix} \text{}{\tau(\mathrm{Pa})} & \text{50} & \text{100} & \text{150} & \text{163} & \text{171} & \text{170} & \text{202} & \text{246} & \text{349} & \text{444}\\ \text{U (m/s)} & \text{0} & \text{0} & \text{0} & \text{0.005} & \text{0.01} & \text{0.025} & \text{0.05} & \text{0.1} & \text{0.2} & \text{0.3}\\ \end{matrix}$$

For the following two-dimensional partial differential equations, find the dispersion relation: $(\mathrm{a})\ \frac{\partial^{2} u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right); (\mathrm{b})\ \frac{\partial^{2} u}{\partial t^{2}}=c_{1}^{2} \frac{\partial^{2} u}{\partial x^{2}}+c_{2}^{2} \frac{\partial^{2} u}{\partial y^{2}}.$

Compute the convolution of each of the following pairs of functions. $\sin a t, \sin b t, a \neq b$

Show that $\int_{e}^{\infty} \frac{d x}{x(\ln x)^{p}}$ converges if p>1 and diverges if $p \leq 1$.

Determine whether each improper integral converges or diverges. If it converges, find its value. $\int_{-\infty}^{1} \frac{x d x}{\sqrt{2-x}}$

Plot the points and find the slope of the line passing through the points. (-1, 5), (2, -3).

An object of mass m is projected vertically downward with initial velocity $V_0$ in a medium offering resistance proportional to the square root of the magnitude of the velocity. (a) Find a relation between the velocity V and the time t if the drag force equals $c \sqrt{V}$ (b) Find the terminal velocity of the object.

Find each definite integral. $\int_{0}^{2} \frac{x^{2}}{\sqrt{9+x^{2}}} d x$

Explain how to determine whether two nonzero vectors u and v are parallel.

These exercises are grouped randomly so you will have to decide which techniques to use. $\int(\sin x+\cos x)^{2} d x$

Show that the average rate of change of the function $x^{3}$ over the interval $[a, b]$, where $0<a<b$, is equal to the instantaneous rate of change of $x^{-3}$ at $x=\sqrt{\left(a^{2}+a b+b^{2}\right) / 3}$. Is this point to the left or to the right of the midpoint $(a+b) / 2$ of the interval [a, b]?

Find the standard equation of the sphere with the given characteristics. Center: (7, 1, −2); Radius: 1