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.

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

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?

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. $y=\left(x^{2}+1\right)^{x}$

Prove the following two identities both algebraically and by interpreting their meaning combinatorially. a.

$$\left(\begin{array}{l} n \\ r \end{array}\right)=\left(\begin{array}{l} n \\ n-r \end{array}\right)$$

b.

$$\left(\begin{array}{l} n \\ r \end{array}\right)=\left(\begin{array}{l} n-1 \\ r-1 \end{array}\right)+\left(\begin{array}{c} n-1 \\ r \end{array}\right)$$

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 $A=P C P^{-1}$.

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

Find the upper and lower central series for $A_{4}$ and $S_{4}$.

A sample of sulfur weighing 0.210 g was dissolved in 17.8 g of carbon disulfide,$\mathrm{CS}_{2}\left(\mathrm{K}_{\mathrm{b}}=2.43^{\circ} \mathrm{C} / \mathrm{m}\right)$. If the boiling point elevation was 0.107 $^{\circ} \mathrm{C}$, what is the formula of a sulfur molecule in carbon disulfide?

Find the absolute maximum value and the absolute minimum value, if any, of the function. $h(t)=8 t-\frac{1}{t^{2}}$ on [1, 3]

Let $f(x)=x^{2}+a x+b$. Determine the constants a and b such that f has a relative minimum at x=2 and the relative minimum value is 7.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $f(x)=3^{x}$, then $f^{\prime}(x)=x \cdot 3^{x-1}$.

A subsidiary of Elektra Electronics manufactures graphing calculators. Management determines that the daily cost C(x) (in dollars) of producing these calculators is $C(x)=0.0001 x^{3}-0.08 x^{2}+40 x+5000$ where x is the number of calculators produced. Find the inflection point of the function C, and interpret your result.

Find both the parametric and the vector equations of the following lines. The line through (-2,4,3) parallel to the x-axis.