Show how the following transformations may be accomplished in good yield. You may use any additional reagents that are needed.

Starting with benzene and using any other necessary reagents of your choice, design a synthesis for anisole (methoxybenzene).

Show how ethyl alcohol could be used to prepare (a) $\mathrm{CH}_{3} \mathrm{CN}$ and (b) $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CN}$. Along with ethyl alcohol you may use any necessary inorganic reagents.

How do the relative proportions of the cranium and face of a fetus compare with those of an adult skull?

There a linear transformation T(x) is defined that gives the costs associated with a production vector x. Determine $T^{−1}$, and use it to find the production level for each type of player that will result in the given costs. If the given costs are not possible, explain why. Labor = $1148, Materials =$1076, and Overhead = $1452.

Show how you would accomplish the following synthetic transformations. You may use any necessary reagents. (a) N-ethylbenzamide $\longrightarrow$ benzylethylamine. (b) ethyl benzoate $\longrightarrow$ N-ethybenzamide. (c) pyrrolidine $\longrightarrow$ N-acetylpyrrolidine (d) $\gamma$-aminobutrric acid $\longrightarrow$ pyrrolidine

Write a series of equations showing how you could prepare octane from acetylene and any necessary organic and inorganic reagents.

Show how each transformation may be accomplished by using a nitrile as an intermediate. You may use any necessary reagents. (a) hexan-1-ol$\longrightarrow$ heptan-1-amine (b) cyclohexanecarboxamide$\longrightarrow$ cyclohexyl ethyl ketone (c) octan-1-ol$\longrightarrow$decan-2-one

Suggest an efficient synthesis of trans-2-heptene from propyne and any necessary organic or inorganic reagents.

Determine if the given matrix is positive definite.

$$A=\left[\begin{array}{rrrr} 1 & -2 & 2 & 1 \\ -2 & 5 & -1 & -3 \\ 2 & -1 & 14 & -1 \\ 1 & -3 & -1 & 3 \end{array}\right]$$

The pmf for X=the number of major defects on a randomly selected appliance of a certain type is

$$\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline p(x) & .08 & .15 & .45 & .27 & .05 \end{array}$$

Compute the following: a. E(X), b. V(X) directly from the definition, c. The standard deviation of X, d. V(X) using the shortcut formula.

Show how you would accomplish the following syntheses using amides as intermediates. You may use any necessary reagents. (a) benzoic acid$\longrightarrow$benzyldimethylamine (b) pyrrolidine$\longrightarrow$ N-ethylpyrrolidine (c) cyclopentanecarboxylic acid $\longrightarrow$ cyclopentanecarbonitrile

Write the prime factorization of each number. Use exponents for repeated factors. 105

Show that $f^{\prime}(0)=f^{\prime \prime}(0)=0$ in the example.