Apply the Root Test to determine convergence or divergence, or state that the Root Test is inconclusive.

$$\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 } { 4 n } \right) ^ { n }$$

Apply the Root Test to determine convergence or divergence, or state that the Root Test is inconclusive.

$$\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n } }$$

Evaluate the function when x = -2, 0, and 3.

$$h ( x ) = \frac { 3 } { 4 } x - 6$$

Find the maximum and minimum points on the graph of $x^{2}+4 y^{2}-2 x-16 y+13=0$, (a) without using calculus. (b) using calculus.

Lithosphere is created at boundaries and destroyed at boundaries. (a) convergent; divergent, (b) divergent; convergent, (c) divergent; transform, (d) convergent; transform.

Convert the given Celsius temperature to its equivalent temperature on the Fahrenheit scale. Where appropriate, round to the nearest tenth of a degree.

$$- 5 ^ { \circ } \mathrm { C }$$

Last year's records of auto accidents occurring on a given section of highway were classified according to whether the resulting damage was $1000 or more and to whether a physical injury resulted from the accident. The data follows:

Under$1000: Number of Accidents 32 and Number Involving injuries 10 $1000 or more: Number of Accidents 41 and Number Involving Injuries 23

a. Estimate the true proportion of accidents involving injuries when the damage was$1000 or more for similar sections of highway and find the margin of error. b. Estimate the true difference in the proportion of accidents involving injuries for accidents with damage under $1000 and those with damage of$1000 or more. Use a 95% confidence interval.

Two systems $\mathrm { K }$ and $\mathrm { K } ^ { \prime }$ synchronize their clocks at $t =$ $t ^ { \prime } = 0$ when their origins are aligned as system $\mathrm { K } ^ { \prime }$ passes by system $\mathrm { K }$ along the $x$ axis at relative speed $0.8 \: \mathrm { c }$ . At time $t = 3 \: \mathrm { ns } ,$ Frank $($ in system $\mathrm { K } )$ shoots a proton gun having proton speeds of $0.98 \: c$ along his $x$ axis. The protons leave the gun at $x = 1 \: \mathrm { m }$ and arrive at a target $120 \: \mathrm { m }$ away. Determine the event coordinates $( x , t )$ of the gun firing and of the protons arriving as measured by observers in both systems $\mathrm { K }$ and $\mathrm { K } ^ { \prime }$ .

Two charged particles are placed near the origin of an xyz coordinate system. Particle 1 carries charge +q and is on the x axis at x = +d. Particle 2 carries an unknown charge and is at an unknown location. The magnitude of the electric field at the origin is $q / \left( 2 \pi \epsilon _ { 0 } d ^ { 2 } \right)$, and the electrostatic potential at the origin, relative to V = 0 at infinity, is $+ 3 q / \left( 4 \pi \epsilon _ { 0 } d \right)$. (a) If particle 2 is on the x axis, what are its charge and location? (b) If particle 2 is on the y axis, what are its charge and location? Determine all possible solutions.

Solve each system by the substitution method . Be sure to check all proposed solutions.

$$\left\{ \begin{array} { r } { x + 8 y = 6 } \\ { 2 x + 4 y = - 3 } \end{array} \right.$$

The radius of a sphere is doubled. What happens to the ratio of the volume of the sphere to the surface area of the sphere? (F) It remains the same. (G) It is doubled. (H) It is increased by a factor of 4. (J) It is increased by a factor of 8

A thin sheet of plastic is uniform and has the shape of an equilateral triangle. Consider two axes for rotation. Both are perpendicular to the plane of the triangle, axis A passing through the center of the triangle and axis B passing through one corner. If the angular speed

$$\omega$$

about each axis is the same, for which axis does the triangle have the greater rotational kinetic energy? Explain.

The distance from Potsdam to larger markets and limited air service have hindered the town in attracting new industry. Air Express, a major overnight delivery service, is considering establishing a regional distribution center in Potsdam. But Air Express will not establish the center unless the length of the runway at the local airport is increased. Another candidate for new development is Diagnostic Research, Inc. (DRI), a leading producer of medical testing equipment. DRI is considering building a new manufacturing plant. Increasing the length of the runway is not a requirement for DRI, but the planning commission feels that doing so will help convince DRI to locate their new plant in Potsdam. Assuming that the town lengthens the runway, the Potsdam planning commission believes that the probabilities shown in the following table are applicable.

$$\begin{matrix} \text{ } & \text{DRI Plant} & \text{No DRI Plant}\\ \text{Air Express Center} & \text{.30} & \text{.10}\\ \text{No Air Express Center} & \text{.40} & \text{.20}\\ \end{matrix}$$

For instance, the probability that Air Express will establish a distribution center and DRI will build a plant is .30. The estimated annual revenue to the town, after deducting the cost of lengthening the runway, is as follows:

$$\begin{matrix} \text{ } & \text{DRI Plant} & \text{No DRI Plant}\\ \text{Air Express Center} & \text{\$600,000} & \text{\$150,000}\\ \text{No Air Express Center} & \text{\$250,000} & \text{-\$200,000}\\ \end{matrix}$$

If the runway expansion project is not conducted, the planning commission assesses the probability DRI will locate their new plant in Potsdam at .6; in this case, the estimated annual revenue to the town will be $450,000. If the runway expansion project is not conducted and DRI does not locate in Potsdam, the annual revenue will be$0 since no cost will have been incurred and no revenues will be forthcoming. a. What is the decision to be made, what is the chance event, and what is the consequence? b. Compute the expected annual revenue associated with the decision alternative to lengthen the runway. c. Compute the expected annual revenue associated with the decision alternative to not lengthen the runway. d. Should the town elect to lengthen the runway? Explain. e. Suppose that the probabilities associated with lengthening the runway were as follows:

$$\begin{matrix} \text{ } & \text{DRI Plant} & \text{No DRI Plant}\\ \text{Air Express Center} & \text{.40} & \text{.10}\\ \text{No Air Express Center} & \text{.30} & \text{.20}\\ \end{matrix}$$

What effect, if any, would this change in the probabilities have on the recommended decision?

Find the exact value of each expression. Do not use a calculator.

$$\sin ^ { - 1 } \left( \cos \frac { 2 \pi } { 3 } \right)$$