Combine the four sentences into an effective topic sentence, or write a topic sentence accompanied by one or more supporting details.
a. Ernie sells hot dogs. b. He is weird, yet lovable. c. Everyone in town knows and likes him. d. He is part of what makes our town interesting.

Show that Lagrange's method automatically yields the correct equations of motion for a particle moving in a plane in a rotating coordinate system Oxy. (Hint: $T=\frac{1}{2} m \mathbf{v} \cdot \mathbf{v}$, where $\mathbf{v}=\mathbf{i}(\dot{x}-\omega y)+\mathbf{j}(\dot{y}+\omega x), \text { and } F_{x}=-\partial \mathrm{V} / \partial x, F_{y}=-\partial \mathrm{V} / \partial y$.)

The partial pressure of carbon dioxide varies with seasons. Would you expect the partial pressure in the Northern Hemisphere to be higher in the summer or winter? Explain.

The partial pressure of carbon dioxide varies with seasons. Would you expect the partial pressure in the northern hemisphere to be higher in the summer or winter? Explain.

A different taxi service charges a pick-up fee of $4 plus a charge of$1.75 per mile driven. Write an equation for this function, and identify the slope and the y-intercept.

Hot air at $80^{\circ} \mathrm{C}$ is blown over a $2-m \times 4-m$ flat surface at $30^{\circ} \mathrm{C}.$ If the average convection heat transfer coefficient is $55 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K},$ determine the rate of heat transfer from the air to the plate, in kW.

Identify the type of curve for each equation, and then view it on a calculator.

$$4(y^2 + 6y + 1) = x(x - 4) - 24$$

Let T = g(x, y) be the temperature at the point (x, y) on the ellipse $x = 2 \sqrt { 2 } \cos t , \quad y = \sqrt { 2 } \sin t , \quad 0 \leq t \leq 2 \pi,$ and suppose that

$$\frac { \partial T } { \partial x } = y , \quad \frac { \partial T } { \partial y } = x.$$

Suppose that T = xy - 2. Find the maximum and minimum values of T on the ellipse.

Write the rational function given as a partial sum of partial fractions.

$$\frac{4 x^{3}+4 x^{2}+x-1}{x^{2}(x+1)^{2}}$$

Find a formal solution to the given initialboundary value problem.

$$\begin{aligned} &\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}, \quad 0<x<\pi, \quad t>0 \\ &u(0, t)=0, \quad u(\pi, t)=3 \pi, \quad t>0 \\ &u(x, 0)=0, \quad 0<x<\pi \end{aligned}$$

A hot-air balloon rises vertically from the ground so that its height after t sec is $h=\frac{1}{2} t^{2}+\frac{1}{2} t \mathrm{ft}(0 \leq t \leq 60)$

What is the velocity of the balloon at the end of 40 sec?

Given the boundary-value problem

$$\begin{array}{ll}{\frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}},} & {0<x<1, \quad 0<t<1} \\ {u(0, t)=0,} & {u(1, t)=0, \quad 0 \leq t \leq 1}\end{array}$$

$u(x, 0)=x(1-x),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0, \quad 0 \leq x \leq 1$ use $h=k=\frac{1}{5}$ in equation (6) to compute the values of $u_{i, 1}$ by hand.

Find the indicated partial sum of the series. $\sum_{i=1}^{\infty} \frac{2}{3^{i}}$ Fifth partial sum

Compare and contrast rotation and revolution from the point of view of angular motion. Then give an astronomical example of each.