A coin is tossed and a number cube is rolled. What is the probability of heads and a number greater than three?

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $\int \frac { d x } { \sqrt { 4 - x ^ { 2 } } }$

Show that the inverse of

$$A=\left[\begin{array}{rr} a&b \\ c&d \end{array}\right]$$

is given by the formula

$$A^{-1}=\left[\begin{array}{rr} \dfrac d{\Delta}& \dfrac {-b}{\Delta} \\\\ \dfrac {-c}{\Delta}&\dfrac {a}{\Delta} \end{array}\right]$$

where $\Delta =ad-bc \neq 0$. The number $\Delta$ is called the determinant of A.

Find the limit by interpreting the expression as an appropriate derivative. (a)

$$\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 3 x ) } { x }$$

, (b)

$$\lim _ { x \rightarrow 0 } \frac { \ln ( 1 - 5 x ) } { x }$$

For the characteristic function

$$\begin{gather*} s^3 + 5s^2 + 9s + 5 \end{gather*}$$

, use Routh’s stability criterion to determine the number of roots with positive real parts. Confirm your answers by using a computer to find the roots.

Independence of events is not always obvious. Toss two balanced coins independently. The four possible combinations of heads and tails in order each have probability 0.25. The events A = head on the first toss B = both tosses have the same outcome may seem intuitively related. Show that P(B|A) = P(B), so that A and B are in fact independent.

The maintenance cost for furnaces at a copper smelting plant have been constant at $140,000 per year for the past 5 years. If the interest rate was 8% per year for the first 3 years and then it increased to 10% in years 4 and 5, what is the equivalent future worth (in year 5) of the maintenance cost? Show hand and spreadsheet solutions.

Vocabulary Types of stores Words to talk about jewelry Other gifts Words to talk about the past

At a certain depth in an incompressible liquid, the absolute pressure is p. At twice this depth, will the absolute pressure be equal to 2p, greater than 2p, or less than 2p? Justify your answer.

Concurso de diseno. Una compania de coches quiere incorporar las ideas de los jovenes en una serie nueva de coches. Patrocinan un concurso en el que grupos de dos o tres estudiantes disenan un coche nuevo. El grupo que gane tendra la oportunidad de trabajar con la compania para crear un prototipo real. En grupos de tres, disenen un coche para el concurso. Deben explicar os rasgos mas importantes e indicar que tipo de persona va a querer comprarlo y por que. Consideren los siguientes aspectos: -el diseno exterior: apariencia, tamano y forma -el diseno interior: comodidad, rasgos para atraer al tipo de persona al que se le vendera -el rendimiento: las habilidades del coche, como se desplaza -la eficiencia energetica: que combustble se usa y cuanto se requiere para accionar el motor

Graph the linear system. Then use the graph to tell whether the linear system has one solution, no solution, or infinitely many solutions.

$$\begin{array} { l } { x - y = 1 } \\ { x - y = 6 } \end{array}$$

Explain the significance of: James K. Polk, manifest destiny, Zachary Taylor, Bear Flag Revolt, Winfield Scott, Treaty of Guadalupe Hidalgo, Mexican Cession.

Rapper, singer, and actress Queen Latifah discovered Giovanni's poetry at age 14. "Nikki's poems struck me," Latifah explains. "I could feel her. I liked how some of the things she wrote were so clever and cool. I liked how she threw a little bit of rhythm around. All her poetry seemed to be real and to have love in it." After reading "Luxury" and "Kidnap Poem," do you agree or disagree with this description? Explain, citing evidence from both poems.

Predict what might happen if a pregnant woman does not include enough folic acid in her diet.