Suppose that a radioactive substance decays according to the model $N=N_{0} e^{-\lambda t}$. (a) Show that affer a period of $T_{\lambda}=1 / \lambda$, the material has decreased to $e^{-1}$ of its original value. $T_{\lambda}$ is called the time constant and it is defined by this property. (b) A certain radioactive substance has a half-life of 12 hours. Compute the time constant for this substance. (c) If there are originally 1000 mg of this radioactive substance present, plot the amount of substance remaining over four time periods $T_{\lambda}$.

After rounding the final turn in the bell lap, two runners emerge ahead of the pack. When Runner A. is 200 ft from the finish line, his speed is 22 ft/sec, a speed that he maintains until he crosses the line. At that instant of time, Runner B, who is 20 ft behind Runner A and running at a speed of 20 ft/sec, begins to sprint. Assuming that Runner B sprints with a constant acceleration, what minimum acceleration will enable him to cross the finish line ahead of Runner A?

Show that the given signal is a solution of the difference equation. Then find the general solution of the difference equation. $y_{k}=k-2 ; y_{k+2}-4 y_{k}=8-3 k$

A drug is carried into an organ of volume V $cm^3$ by a liquid that enters the organ at the rate of a $\mathrm{cm}^{3} / \mathrm{sec}$ and leaves it at the rate of b $\mathrm{cm}^{3} / \mathrm{sec}$. The concentration of the drug in the liquid entering the organ is c $\mathrm{g} / \mathrm{cm}^{3}$. If the concentration of the drug in the organ at time t is increasing at the rate of $x^{\prime}(t)=\frac{1}{V}\left(a c-b x_{0}\right) e^{-b t v}$ $\mathrm{g} / \mathrm{cm}^{3} / \mathrm{sec}$ and the concentration of the drug in the organ initially is $x_{0} \mathrm{g} / \mathrm{cm}^{3}$, show that the concentration of the drug in the organ at time t is given by $x(t)=\frac{a c}{b}+\left(x_{0}-\frac{a c}{b}\right) e^{-b c v}$

Find the indefinite integral. $\int\left(1+u+u^{2}\right) d u$

Find the indefinite integral. $\int\left(1+x+e^{x}\right) d x$

Use the data in the table, which shows the average annual salaries (both in thousands of dollars) for librarians and postsecondary library science teachers in the United States for 12 years.

$$\begin{matrix} \text{Librarians, x} & \text{Library science teachers, y}\\ \text{49.1} & \text{56.6}\\ \text{50.9} & \text{57.6}\\ \text{52.9} & \text{59.7}\\ \text{54.7} & \text{61.6}\\ \text{55.7} & \text{64.3}\\ \text{56.4} & \text{67.0}\\ \text{57.0} & \text{70.0}\\ \text{57.2} & \text{70.8}\\ \text{57.6} & \text{73.3}\\ \text{58.1} & \text{72.4}\\ \text{58.9} & \text{73.0}\\ \text{59.9} & \text{72.3}\\ \end{matrix}$$

Construct a scatter plot for the data. Do the data appear to have a positive linear correlation, a negative linear correlation, or no linear correlation? Explain.

Find the density function of $Y=e^{Z}$, where $Z \sim N\left(\mu, \sigma^{2}\right)$. This is called the lognormal density, since log Y is normally distributed.

Suppose that the probability of living to be older than 70 is .6 and the probability of living to be older than 80 is .2. If a person reaches her 70th birthday, what is the probability that she will celebrate her 80th?

In the children's game rock, paper, scissors, two players simultaneously extend one hand in the form of a fist (rock), a flat palm (paper), or two fingers extended (scissors). If they both make the same choice, the game is a tie. Otherwise, rock beats scissors (because rock can crush scissors), scissors beats paper (because scissors can cut paper), and paper beats rock (because paper can cover rock). Whoever wins gains one point; in the case of a tie, no one gains any points. (a) Find the optimum strategy for each player and the value of the game. (b) In hindsight, how could you have guessed the answer to part (a) without doing any work?

Show that a group of order 24 with no element of order 6 is isomorphic to $S_{4}$.

The number of saltwater crocodiles in a certain area of northern Australia is given by $P(t)=\frac{300 e^{-0.024 t}}{5 e^{-0.024 t}+1}$ a. How many crocodiles were in the population initially? b. Show that $\lim _{t \rightarrow \infty} P(t)=0$. c. Plot the graph of P in the viewing window $[0,200] \times [0,70]$.

Use logarithmic differentiation to find the derivative of the function. $y=\left(x^{2}+1\right)^{x}$

Find an invertible matrix P and a matrix C of the form

$$\left[\begin{array}{rr} a & -b \\ b & a \end{array}\right]$$

such that the given matrix has the form $A=P C P^{-1}$.

$$\left[\begin{array}{rr} 4 & -2 \\ 1 & 6 \end{array}\right]$$