An electromagnetic wave propagating in vacuum has electric and magnetic fields given by $\vec{E}(\vec{x}, t)=\vec{E}_0 \cos (\vec{k} \cdot \vec{x}-\omega t)$ and $\vec{B}(\vec{x}, t)=\vec{B}_0 \cos (\vec{k} \cdot \vec{x}-\omega t)$, where $\vec{B}_0$ is given by $\vec{B}_0=\vec{k} \times \vec{E}_0 / \omega$ and the wave vector $\vec{k}$ is perpendicular to both $\vec{E}_0$ and $\vec{B}_0$. The magnitude of $\vec{k}$ and the angular frequency $\omega$ satisfy the dispersion relation, $\omega /|\vec{k}|=\left(\mu_0 \epsilon_0\right)^{-1 / 2}$, where $\mu_0$ and $\epsilon_0$ are the permeability and permittivity of free space, respectively. Such a wave transports energy in both its electric and magnetic fields. Calculate the ratio of the energy densities of the magnetic and electric fields, $u_B / u_E$, in this wave. Explain.

a. Approximate the Cauchy constants $A$ and $B$ for crown and flint glasses, using data for the $C$ and $F$ Fraunhofer lines from Table 1 . Using these constants and the Cauchy relation approximated by two terms, calculate the refractive index of the $D$ Fraunhofer line for each case. Compare your answers with the values given in the table. b. Calculate the dispersion in the vicinity of the Fraunhofer $D$ line for each glass, using the Cauchy relation. c. Calculate the chromatic resolving power of crown and flint prisms in the vicinity of the Fraunhofer $D$ line, if each prism base is $75 \mathrm{~mm}$ in length. Also calculate the minimum resolvable wavelength interval in this region.

Prof. Hardtack gave four Friday quizzes last semester in his 10-student senior tax accounting class. (a) Find the sample mean, standard deviation, and coefficient of variation for each quiz. (b) How do these data sets differ in terms of central tendency and dispersion? (c) Briefly describe and compare student performance on each quiz.

Quiz 1: 60, 60, 60, 60, 71, 73, 74, 75, 88, 99

Quiz 2: 65, 65, 65, 65, 70, 74, 79, 79, 79, 79

Quiz 3: 66, 67, 70, 71, 72, 72, 74, 74, 95, 99

Quiz 4: 10, 49, 70, 80, 85, 88, 90, 93, 97, 98

Consider a general dispersion relation of the form $E=\alpha p^{s}$, where p is the momentum and $\alpha \text { and } p$ are constants. Using the result of the previous question, show that the density of states as a function of energy is $g(E) \mathrm{d} E=\frac{g V D \pi^{D / 2}}{h^{D} \alpha^{D / s} s \Gamma\left(\frac{D}{2}+1\right)} E^{\frac{D}{s}-1} \mathrm{d} E$. Hence show that the single-particle partition function takes the form $Z_{1}=\frac{V}{\lambda^{D}}$, where $\lambda$ is given by $\lambda=\frac{h}{\pi^{1 / 2}}\left(\frac{\alpha}{k_{\mathrm{B}} T}\right)^{1 / s}\left[\frac{\Gamma\left(\frac{D}{2}+1\right)}{\Gamma\left(\frac{D}{s}+1\right)}\right]^{1 / D}$. Show that this result for three dimensions (D = 3) agrees with (i) the non-relativistic case when s = 2 and (ii) the ultrarelativistic case when s = 1.

Two waves are described by the wave functions

$$\begin{array}{l}y_{1}(x, t)=5.00 \sin (2.00 x-10.0 t) \\ y_{2}(x, t)=10.0 \cos (2.00 x-10.0 t)\end{array}$$

where x, $y_{1},$ and $y_{2}$ are in meters and t is in seconds. (a) Show that the wave resulting from their superposition can be expressed as a single sine function. (b) Determine the amplitude and phase angle for this sinusoidal wave.

A. H. Pfund's method for measuring the index of refraction of glass is illustrated. One face of a slab of thickness $t$ is painted white, and a small hole scraped clear at point $P$ serves as a source of diverging rays when the slab is illuminated from below. Ray $P B B^{\prime}$ strikes the clear surface at the critical angle and is totally reflected, as are rays such as $P C C^{\prime}$. Rays such as $P A A^{\prime}$ emerge from the clear surface. On the painted surface there appears a dark circle of diameter $d$, surrounded by an illuminated region, or halo. (c) If white light is used, the critical angle depends on color caused by dispersion. Is the inner edge of the white halo tinged with red light or violet light? Explain.

A student wanted to know whether Centrum vitamins dissolve fast er than the corresponding generic brand. The student used vinegar as a proxy for stomach acid and measured the time (in minutes) it took for a vitamin to completely dissolve. The results are shown next.

$$\begin{matrix} \text{Centrum} & \text{ } & \text{ } & \text{ } & \text{ } & \text{Generic Brand}\\ \text{2.73} & \text{3.07} & \text{3.30} & \text{3.35} & \text{3.12} & \text{6.57} & \text{6.23} & \text{6.17} & \text{7.17} & \text{5.77}\\ \text{2.95} & \text{2.15} & \text{2.67} & \text{2.80} & \text{2.25} & \text{6.73} & \text{5.78} & \text{5.38} & \text{5.25} & \text{5.55}\\ \text{2.60} & \text{2.57} & \text{4.02} & \text{3.02} & \text{2.15} & \text{5.50} & \text{6.50} & \text{7.42} & \text{6.47} & \text{6.3}\\ \text{3.03} & \text{3.53} & \text{2.63} & \text{2.30} & \text{2.73} & \text{6.33} & \text{7.42} & \text{5.57} & \text{6.35} & \text{5.92}\\ \text{3.92} & \text{2.38} & \text{3.25} & \text{4.00} & \text{3.63} & \text{5.35} & \text{7.25} & \text{7.58} & \text{6.5} & \text{4.97}\\ \text{3.02} & \text{4.17} & \text{4.33} & \text{3.85} & \text{2.23} & \text{7.13} & \text{5.98} & \text{6.6} & \text{5.03} & \text{7.18}\\ \end{matrix}$$

(a) Draw side-by-side boxplot s for each vitamin type. (b) Which vitamin type has more dispersion? (c) Which vitamin type appears to dissolve faster?

Explica con tus palabras estos términos y expresiones que aparecen en el texto.

• vida sedentaria

Si un cáncer es diagnosticado a tiempo, puede ser eliminado por diversos medios (normalmente mediante una intervención quirúrgica ${ }^6$ ). A continuación, los médicos prescriben un tratamiento con unos fármacos especiales que impiden la actividad de las células cancerosas y frenan ${ }^7$ su dispersión. Este tratamiento se denomina quimioterapia y tiene el inconveniente de que es muy agresivo con el organismo. En algunos casos se realiza un tratamiento con rayos (radioterapia) como complemento del anterior.

Fuente: La enciclopedia del estudiante. Ciencias de la vida. Santillana

• cells - tissues - mass - se produce - temprana - operación - slow

Prof. Hardtack gave four Friday quizzes last semester in his 10-student senior tax accounting class.

(a) Using Excel, find the sample mean, standard deviation, and coefficient of variation for each quiz.

(b) How do these data sets differ in terms of central tendency and dispersion?

(c) Briefly describe and compare student performance on each quiz.

Quiz 1: 60, 60, 60, 60, 71, 73, 74, 75, 88, 99

Quiz 2: 65, 65, 65, 65, 70, 74, 79, 79, 79, 79

Quiz 3: 66, 67, 70, 71, 72, 72, 74, 74, 95, 99

Quiz 4: 10, 49, 70, 80, 85, 88, 90, 93, 97, 98

"The wave function $y(x, t)$ for a certain standing wave on a string that is fixed at both ends is given by $y(x, t)=4.20 \sin (0.200 x) \cos (300 t)$, where $y$ and $x$ are in centimeters and $t$ is in seconds. A standing wave can be considered as the superposition of two traveling waves. $(c)$ If the string is vibrating in its fourth harmonic, how long is it? "

Monthly stock prices (in $) for two competing firms are as follows.$

$$\begin{array}{|l|c|c|} \hline \text { Month } & \text { Firm A } & \text { Firm B } \\ \hline \text { January } & 28 & 21 \\ \hline \text { February } & 31 & 24 \\ \hline \text { March } & 32 & 24 \\ \hline \text { April } & 35 & 27 \\ \hline \text { May } & 34 & 25 \\ \hline \text { June } & 28 & 20 \\ \hline \end{array}$$

$

a. Calculate the sample mean, the sample variance, and the sample standard deviation for each firm's stock price.

b. Which firm had the higher average stock price over the time period?

c. Which firm's stock price had greater variability as measured by the standard deviation? Which firm's stock price had the greater relative dispersion?

Devise an algorithm for constructing the spanning forest of a graph based on depth-first searching.

The index of refraction in flint glass is 1.66 for blue light and 1.61 for red light, with the values for other colors between these limits. A ray of white light traveling from left to right through a slab of flint glass undergoes dispersion as it exits the slab and enters the air. If the exiting ray strikes the right surface of the slab at an angle of $30.0 ^ { \circ }$ from the normal to the surface, what is the angle of refraction (a) for the blue light and (b) for the red light? (c) A flat opaque screen is placed in the path of the refracted rays such that the red ray strikes perpendicular to the screen. If the distance along the refracted red ray from slab to screen is 0.50 m, what is the distance between the point of red light and the point of blue light in the rainbow produced on the screen? (Use the small-angle approximation.)

This table shows the fertility rate (children born per woman) in 191 world nations. (a) From the grouped data, calculate the mean, standard deviation, and coefficient of variation for each year. Show your calculations clearly in a worksheet. (b) Write a concise comparison of central tendency and dispersion for fertility rates in the two years. (c) What additional information would you have gained by having the raw data? (d) What benefit arises from having only a summary table?

$$\begin{array}{cccr} \hline \text { From } & \text { To } & 1990 & \mathbf{2 0 0 0} \\ \hline 1.00 & 2.00 & 39 & 54 \\ 2.00 & 3.00 & 35 & 44 \\ 3.00 & 4.00 & 27 & 23 \\ 4.00 & 5.00 & 26 & 22 \\ 5.00 & 6.00 & 24 & 26 \\ 6.00 & 7.00 & 30 & 16 \\ 7.00 & 8.00 & 9 & 5 \\ 8.00 & 9.00 & 1 & 1 \\ & \text { Total } & 191 & 191 \\ \hline \end{array}$$