Find a polar equation of the conic with its focus at the pole. Parabola; vertex: $(2, \pi / 2)$

In Exercises $1$ through $10$, the probability density function for a continuous random variable $X$ is given. In each case, find the expected value $E(X)$ and the variance $\operatorname{Var}(X)$ of $X$.

$$f(x)= \begin{cases}\frac{3}{32}\left(4 x-x^2\right) & \text { if } 0 \leq x \leq 4 \\ 0 & \text { otherwise }\end{cases}$$

Repeat the above problem by replacing air by oil with a specific gravity of $0.69$.

Explain why the quadratic formula, in most cases, is easier to use to solve a quadratic equation than is the method of completing the square.

For the given differential equation, (a) Determine the complementary solution. (b) List the form of particular solution prescribed by the method of undetermined coefficients; you need not evaluate the constants in the assumed form. [Hint: In Exercises 20 and 22, rewrite the hyperbolic functions in terms of exponential functions. In Exercise 21, use trigonometric identities.]

$$y^{\prime \prime}+4 y=2 \sinh t \cosh t+\cosh ^2 t$$

Dry air is inhaled at a rate of 10 liter/min through a trachea with a diameter of 20 mm and a length of 125 mm. The inner surface of the trachea is at a normal body temperature of $37^{\circ} \mathrm{C}$ and may be assumed to be saturated with water. (a) Assuming steady, fully developed flow in the trachea, estimate the mass transfer convection coefficient. (b) Estimate the daily water loss (liter/day) associated with evaporation in the trachea.

Calculate the de Broglie wavelengths of the following items: a) an electron with a kinetic energy of $3 \mathrm{eV}$ b) a proton with a kinetic energy of $7 \mathrm{MeV}$ c) a buckyball $\left(\mathrm{C}_{60}\right)$ with a speed of $200 \mathrm{~m} / \mathrm{s}$ d) an oxygen molecule at room temperature e) a raindrop f) yourself walking to class In which of the above cases might you expect quantum mechanics to play an important role and why?

Sketch the graph of the polar equation and find the tangents at the pole.

r=2 cos 3 $\theta$

The direction angles $\alpha, \beta$, and $\gamma$ of a vector $\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$ are defined as follows: $\alpha$ is the angle between $\mathbf{v}$ and the positive $x$-axis $(0 \leq \alpha \leq \pi)$. $\beta$ is the angle between $\mathbf{v}$ and the positive $y$-axis $(0 \leq \beta \leq \pi)$. $\gamma$ is the angle between $\mathbf{v}$ and the positive $z$-axis $(0 \leq \gamma \leq \pi)$.

(b). Unit vectors are built from direction cosines Show that if $\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}$ is a unit vector, then $a, b$, and $c$ are the direction cosines of $\mathbf{v}$.

Suppose a firm earns revenue $R(Q)=a Q-b Q^2$ and incurs cost $C(Q)=\alpha Q+\beta Q^2$ as functions of output $Q \geq 0$, where $a, b, \alpha$, and $\beta$ are positive parameters. The firm maximizes profit $\pi(Q)=R(Q)-C(Q)$ subject to the constraint $Q \geq 0$. Solve this one-variable problem by the Kuhn-Tucker method, and find conditions for the constraint to bind at the optimum.

Sketch the general shape of the surface $M: z=a x^{2}+b y^{2}$ in each of the following cases: (a) $a>b>0$ (b) $a>0>b$ (c) $a>b=0$ (d) $a=b=0$

On a toy assembly line, 3% of the toys are found to be defective . The quality control officer predicts that 872 toys will be found defective out of 24,850 toys made. Do you agree with this prediction? Explain

Except for degenerate cases, the equation $A x^{2}+B x y+C y^{2}+D x+E y+F=0$ defines a(n) _____ if $B^{2}-4 A C=0.$

A window air conditioner cools a room at TL = 22$^{\circ} \mathrm{C}$, with a maximum of 1.2 kW power input possible. The room gains 0.6 kW per degree temperature difference to the ambient, and the refrigeration COP is $\beta=0.6 \beta$ Carnot. Find the actual power required on a day when the temperature is 30$^{\circ} \mathrm{C}$ outside.