If a single constant force acts on an object that moves on a straight line, the object’s velocity is a linear function of time. The equation v = vi + at gives its velocity v as a function of time, where a is its constant acceleration. What if velocity is instead a linear function of position? Assume that as a particular object moves through a resistive medium, its speed decreases as described by the equation v = vi − kx, where k is a constant coefficient and x is the position of the object. Find the law describing the total force acting on this object.

Many states face a shortage of fully credentialed teachers. The percentages of teachers who are fully credentialed for each county in California were published in the San Luis Obispo Tribune (July 29, 2001) and are given in the following table: a. Construct a stem-and-leaf display for this data set using stems 7, 8, 9, and 10. Truncate the leaves to a single digit. Comment on the interesting features of the display. b. Construct a stem-and-leaf display using repeated sterns. Are there characteristics of the data set that are easier to see in the plot with repeated stems, or is the general shape of the two displays similar?

$$\begin{matrix} \text{Country} & \text{Number of people (in thousands)}\\ \text{Alameda} & \text{85.1}\\ \text{Alpine} & \text{100.0}\\ \text{Amador} & \text{97.3}\\ \text{Butte} & \text{98.2}\\ \text{Calaveras} & \text{97.3}\\ \text{Colusa} & \text{92.8}\\ \text{Contra Costa} & \text{87.7}\\ \text{Del Norte} & \text{98.8}\\ \text{El Dorado} & \text{96.7}\\ \text{Fresno} & \text{91.2}\\ \text{Glenn} & \text{95.0}\\ \text{Humbolt} & \text{98.6}\\ \text{Imperial} & \text{79.8}\\ \text{Ino} & \text{94.4}\\ \text{Kern} & \text{85.1}\\ \text{Kings} & \text{85.0}\\ \text{Lake} & \text{95.0}\\ \text{Lassen} & \text{89.6}\\ \text{Los Angeles} & \text{74.7}\\ \text{Madera} & \text{91}\\ \text{Marin} & \text{96.8}\\ \text{Mariposa} & \text{95.5}\\ \text{Mendieino} & \text{97.2}\\ \text{Mereed} & \text{87.8}\\ \text{Modoc} & \text{94.6}\\ \text{Mono} & \text{95.6}\\ \text{Monteray} & \text{84.6}\\ \text{Napa} & \text{90.8}\\ \text{Nevada} & \text{93.9}\\ \end{matrix}$$

$$\begin{matrix} \text{Country} & \text{Number of people (in thousands)}\\ \text{Orange} & \text{91.3}\\ \text{Placer} & \text{97.8}\\ \text{Plumas} & \text{95.0}\\ \text{Riverside} & \text{84.5}\\ \text{Sacramento} & \text{95.3}\\ \text{San Benito} & \text{83.5}\\ \text{San Bernardino} & \text{83.1}\\ \text{San Diego} & \text{96.6}\\ \text{San Francisco} & \text{94.4}\\ \text{San oaquin} & \text{86.3}\\ \text{San Luis Obsipo} & \text{98.1}\\ \text{San Mateo} & \text{88.8}\\ \text{Santa Barbara} & \text{95.5}\\ \text{Santa Clara} & \text{84.6}\\ \text{Santa Cruz} & \text{89.6}\\ \text{Shasta} & \text{97.5}\\ \text{Sierra} & \text{88.5}\\ \text{Siskiyou} & \text{97.7}\\ \text{Solano} & \text{87.3}\\ \text{Sonoma} & \text{96.5}\\ \text{Stanislaus} & \text{94.4}\\ \text{Sutter} & \text{89.3}\\ \text{Tehama} & \text{97.3}\\ \text{Trinity} & \text{97.5}\\ \text{Tulare} & \text{87.6}\\ \text{Tuolomne} & \text{98.5}\\ \text{Ventura} & \text{92.1}\\ \text{Yolo} & \text{94.6}\\ \text{Yuba} & \text{91.6}\\ \end{matrix}$$

Use the Comparison Tests for Convergence or Divergence to determine whether each series converges or diverges. $\sum_{k=2}^{\infty} \frac{\sqrt{k}}{k-1}$: by comparing it with $\sum_{k=2}^{\infty} \frac{1}{\sqrt{k}}$

The amount of aluminum contamination (in parts per million) in plastic was determined for a sample of 26 plastic specimens, resulting in the following data (“The Log Normal Distribution for Modeling Quality Data When the Mean Is Near Zero,” Journal of Quality Technology [1990]: 105– 110): Construct a boxplot that shows outliers, and comment on the interesting features of this plot.

$$\begin{array}{ccccccccc}{30} & {30} & {60} & {63} & {70} & {79} & {87} & {90} & {101} \\ {102} & {115} & {118} & {119} & {119} & {120} & {125} & {140} & {145} \\ {172} & {182} & {183} & {191} & {222} & {244} & {291} & {511}\end{array}$$

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A converging-diverging nozzle is attached to a very large tank of air in which the pressure is 150 kPa and the temperature is $35^{\circ} \mathrm{C}$. The nozzle exhausts to the atmosphere where the pressure is 101 kPa. The exit diameter of the nozzle is 2.75 cm. What is the flow rate through the nozzle? Assume the flow is isentropic. At the design condition of the system, the exit Mach number is $M_{e}$=2.0. Find the pressure in the tank (keeping the temperature constant) for this condition. What is the flow rate? What is the throat area?

Stress can affect the physiology and behavior of animals, just as it can with humans (as many of us know all too well). The accompanying data on x = plasma cord-sol concentration (in milligrams of cortisol per milliliter of plasma) and y = oxygen consumption rate (in milligrams per kilogram per hour) for juvenile steelhead after three 2-min disturbances were read from a graph in the paper "Metabolic Cost of Acute Physical Stress in Juvenile Steel-head" (Transactions of the American Fisheries Society [19811: 257-263). The paper also included data for un-stressed fish. a. Is the value of y determined solely by the value of x? Explain your reasoning. b. Construct a scatterplot of the data. c. Does an increase in plasma cortisol concentration ap-pear to be accompanied by a change in oxygen consumption rate? Comment.

$$\begin{matrix} \text{x} & \text{25} & \text{36} & \text{48} & \text{59} & \text{62} & \text{72} & \text{80} & \text{100} & \text{100} & \text{137}\\ \text{y} & \text{155} & \text{184} & \text{180} & \text{220} & \text{280} & \text{163} & \text{230} & \text{222} & \text{241} & \text{350}\\ \end{matrix}$$

To assemble a piece of furniture, a wood peg must be inserted into a predrilled hole. Suppose that the diameter of a randomly selected peg is a random variable with mean 0.25 inch and standard deviation 0.006 inch and that the diameter of a randomly selected hole is a random variable with mean 0.253 inch and standard deviation 0.002 inch. Let $x_1$ = peg diameter, and let $x_2$ = denote hole diameter a. Why would the random variable y, defined as $y = x_2 - x_1$, be of interest to the furniture manufacturer? b. What is the mean value of the random variable y? c. Assuming that $x_1$ and $x_2$ are independent, what is the standard deviation of y? d. Is it reasonable to think that $x_1$ and $x_2$ are independent? Explain. e. Based on your answers to Parts (b) and (c), do you think that finding a peg that is too big to fit in the predrilled hole would be a relatively common or a relatively rare occurrence? Explain.

Sketch the plane curve and find its length over the given interval. $\mathbf{r}(t)=10 \cos t \mathbf{i}+10 \sin t \mathbf{j}$, $[0,2 \pi]$

In the linear space of all real polynomials, determine whether or not (f, g) is an inner product if (f,g) is defined by the formula given. In case (f,g) is not an inner product, indicate which axioms are violated. In (c), $f^{\prime}$ and $g^{\prime}$ denote derivatives. (a) $(f, g)=f(1) g(1)$. (b) $(f, g)=\left|\int_{0}^{1} f(t) g(t) d t\right|$, (c) $(f, g)=\int_{0}^{1} f^{\prime}(t) g^{\prime}(t) d t$, (d) $(f, g)=\left(\int_{0}^{1} f(t) d t\right)\left(\int_{0}^{1} g(t) d t\right)$

Use the Limit Comparison Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{1}{(k+1)(k+2)}$

Consider the vector-valued function $\mathbf{r}(t)=3 t^{2} \mathbf{i}+(t-1) \mathbf{j}+t \mathbf{k}$. Write a vector-valued function u(t) that is the specified transformation of r. A horizontal translation one unit in the direction of the positive x-axis

Use the substitution $x=\pi-u$ to show that $\int_{0}^{\pi} x f(\sin x) d x=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) d x$ for any function f continuous on [0, 1].

At the site of a spill of radioactive iodine, radiation levels were four times the maximum acceptable limit, so an evacuation was ordered. If $R_0$ is the initial radiation level (at t=0) and t is the time in hours, the radiation level R(t), in millirems/hour, is given by $R(t)=R_{0}(0.996)^{t}$. How much total radiation (in millirems) has been emitted by that time?

Find the two roots of $x^{4}-8 x^{2}-x+16=0 \text { in }[1,3]$.