Take the derivative of the previous expression to find an expression for sinh(2x).

For the following exercise, P is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle $\theta$ with a terminal side that passes through point P. Rationalize denominators. $P\left(x, \frac{\sqrt{7}}{3}\right)$, x<0

In a game of poker, five players are each dealt 5 cards from a 52-card deck. How many ways are there to deal the cards?

Find the eigenvalues, and if necessary the corresponding eigenvectors, of A and determine whether A is diagonalizable.

$$A=\left[\begin{array}{llll} 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right]$$

For the following exercise, consider a rocket shot into the air that then returns to Earth. The height of the rocket in meters is given by $h(t)=600+78.4 t-4.9 t^{2}$, where t is measured in seconds. Use the preceding exercise to guess the instantaneous velocity of the rocket at t = 9 sec.

Prove that any finite group has a finite exponent. Give an example of an infinite group with finite exponent. Does a finite group of exponent m always contain an element of order m?

Use the data to (a) find the coefficient of determination $r^2$ and interpret the result, and (b) find the standard error of estimates, and interpret the result. The table shows the combined city and highway fuel efficiency (in miles per gallon gasoline equivalent) and top speeds (in miles per hour) for nine hybrid and electric cars. The regression equation is $\hat{y}=-0.465 x+139.433$

$$\begin{matrix} \text{Fuel efficiency, x} & \text{114} & \text{95} & \text{120} & \text{105} & \text{107} & \text{116} & \text{118} & \text{68} & \text{84}\\ \text{Top speed, y} & \text{80} & \text{103} & \text{78} & \text{85} & \text{92} & \text{88} & \text{92} & \text{105} & \text{101}\\ \end{matrix}$$

For the differential equations, find the indicial polynomial for the singularity at x=0. Then find the recurrence formula for the largest of the roots to the indicial equation. xy''+(1-x)y'-y=0

The number of US. citizens aged 65 years and older from 1900 through 2050 is estimated to be growing at the rate of $R(t)=0.063 t^{2}-0.48 t+3.87$ $(0 \leq t \leq 15)$ million people/decade, where t is measured in decades, with t=0 corresponding to 1900. Show that the average rate of growth of U.S. citizens aged 65 years and older between 2000 and 2050 will be about twice the average rate between 1950 and 2000.

Model each application with a differential equation. All rates are assumed to be with respect to time. The voltage drop across an inductor is proportional to the rate at which the current is changing with respect to time.

Diagonalize the matrices if possible. The real eigenvalues are included below the matrix.

$$\begin{array}{c} {\left[\begin{array}{cc} 2 & 0 & -2 \\ 1 & 3 & 2 \\ 0 & 0 & 3 \end{array}\right]} \\ \lambda=2,3 \end{array}$$

The number of Medicare-certified home-health-care agencies (70% are freestanding, certified home-health-care agencies (70% are freestanding, been declining at the rate of $0.186 e^{-0.02 t}$ $0.186 e^{-0.02 t}$ thousand agencies/year between 1988 (t = 0) and 2002 (t = 14). The number of such agencies stood at 9.3 thousand in 1988. Find an expression giving the number of health-care agencies in year t.

Find the density of cX when X follows a gamma distribution. Show that only $\lambda$ is affected by such a transformation, which justifies calling $\lambda$ a scale parameter.

Find the indefinite integral. $\int 6 d x$