(a) Derive an expression for the angle between the vector representing angular momentum l with z-component $m_{1}=+l$ (that is, its maximum value) and the z-axis. What is this angle for l = 1 and for l = 5? (b) Derive an expression for the angle between the vector representing angular momentum l with z-component $m_{1}=+l$ and the z-axis. What value does this angle take in the limit that l becomes very large? Interpret your result in the light of the correspondence principle.

In the course of a mammal's lifetime, its heart beats about 1 billion times, regardless of the mammal's size or weight. (This excludes humans.) a. An elephant's heart beats approximately 25 times a minute. How many years would you expect an elephant to live? Use scientific notation to calculate your answer. b. A pygmy shrew's heart beats approximately 1150 times a minute. How many years would you expect a pygmy shrew to live? c. If this relationship were true for humans, how many years would you expect a human being with a heart rate of 60 beats per minute to live?

Suppose that a quantity y has an exponential growth model

$$y = y _ { 0 } e ^ { k t }$$

or an exponential decay model

$$y = y _ { 0 } e ^ { - k t }$$

, and it is known that y=

$$y_1$$

if t=

$$t_1$$

. In each case find a formula for k in terms of

$$y_0$$

,

$$y_1$$

, and

$$t_1$$

, assuming that

$$t _ { 1 } \neq 0$$

.

The ideas expressed in the excerpt were most directly a response to the a. decreasing prices of many goods as the real wages of workers increased b. increasing ethnic diversity of an expanding workforce. c. greater concentration of wealth through consolidation of corporations into trusts. d. growing power of political machines in urban areas.

Determine whether matrix

$$\textbf{A}=\begin{bmatrix}4&2&3\\2&1&2\\-1&2&0\end{bmatrix}$$

is diagonalizable. If it is, determine a matrix $\textbf{P}$ that diagonalizes it and compute $\textbf{P}^{-1}\textbf{AP}$. You can obtain $\textbf{P}^{-1}\textbf{AP}$ directly from careful construction of a diagonal matrix with eigenvalues along the diagonal in the proper order.

Find the LCM for each set of numbers. List the multiples. 6, 8, and 4

An unknown three terminal device has leads named b, c, and e. When installed in one particular circuit, measurements indicated that $V_{ec} = –9 \text{ V}$ and $V_{eb} = –0.65 \text{ V}$. (a) Calculate $V_{cb}$. (b) Determine the power dissipated in the b-e junction if the current $I_b$ flowing into the terminal marked b is equal to $1 \; \mu\text{A}$.

What are the largest- and smallest-known values of the mass, luminosity, surface temperature, and diameter of stars (roughly)?

Show that postage of 6 cents or more can be achieved by using only 2-cent and 7-cent stamps.

Simplify the expression.

$$13 ^ { \frac { 1 } { 2 } } \cdot 13 ^ { \frac { 3 } { 2 } }$$

Use a CAS to solve the initial value problem. Plot the solution curves.

$$y^{\prime}=\frac{1}{\sqrt{4-x^{2}}}, \quad y(0)=2$$

Using Newton's Law of Gravitation, if there are two objects exerting the force of gravity on each other and the mass of one of the objects is quadrupled, then the force between them is a. doubled. b. quartered. c. tripled. d. quadrupled.

Find the second derivative of the given function. $f(x)=e^{2 x}+2 e^{-x}$

Water is pumped through a pipe of diameter 15.0 cm from the Colorado River up to Grand Canyon Village, on the rim of the canyon. The river is at 564 m elevation and the village is at 2 096 m. What additional pressure is necessary to deliver this flow? Note: You may assume the free-fall acceleration and the density of air are constant over the given range of elevations.