Suppose that only .10% of all computers of a certain type experience CPU failure during the warranty period. Consider a sample of 10,000 computers. a. What are the expected value and standard deviation of the number of computers in the sample that have the defect? b. What is the (approximate) probability that more than 10 sampled computers have the defect? c. What is the (approximate) probability that no sampled computers have the defect?

A system has four computers. Computer 1 works with a probability of 0.88; computer 2 works with a probability of 0.78; computer 3 works with a probability of 0.92; computer 4 works with a probability of 0.85. Suppose that the operations of the computers are independent of each other.

$\mathbf{(a)}$ Suppose that the system works only when all four computers are working. What is the probability that the system works? $\mathbf{(b)}$ Suppose that the system works only if at least one computer is working. What is the probability that the system works? $\mathbf{(c)}$ Suppose that the system works only if at least three computers are working. What is the probability that the system works?

Differentiate the functions with respect to the independent variable. $f(x)=2^{x+1}$

(a) Explain why we speak of acidic hydrolysis of an ester as acid-catalyzed, but of basic hydrolysis as base-promoted. (b) Soap manufacturers always use base to hydrolyze fats, and never acid. Suggest two reasons that basic hydrolysis is preferred.

Find the derivative with respect to the independent variable. $f(x)=4 \cos x^{2}-2 \cos ^{2} x$

The Beverton–Holt recruitment curve is given by the recursion $N_{t+1}=\frac{R N_{t}}{1+\frac{R-1}{K} N_{t}}$ where R > 1 and K > 0. When $N_{0}>0, \lim _{t \rightarrow \infty} N_{t}=K$ for all values of R > 0. To investigate how R affects the limiting behavior of $N_t,$ find $N_t$ for t = 1, 2, 3, . . . , 10 for K = 100 and $N_{0}=20$ when (a) R = 2, (b) R = 5, and (c) R = 10, and plot $N_t$ as a function of t for the three choices of R in one coordinate system.

Graph the line $N_{t+1}=R N_{t}$ in the $N_{t}-N_{t+1}$ plane for the indicated value of R and locate the points $\left(N_{t}, N_{t+1}\right), t=0, 1,$ and 2, for the given value of $N_{0}.$ $R=\frac{1}{2}, N_{0}=16$

Show how you would synthesize Leu-Gly-Ala-Val-Phe starting with Boc-Ala-Val-Phe

Use the fact that $\sec x=\frac{1}{\cos x}$ to explain why the maximum domain of y=sec x consists of all real numbers except odd integer multiples of $\pi / 2.$

Write the first five terms of the sequence $\left\{a_{n}\right\}, n=0,1,2,3, \ldots,$ and find $\lim _{n \rightarrow \infty} a_{n}.$ $a_{n}=\frac{1}{\sqrt{n+1}}$

Assume that A is a matrix with three rows. Find the matrix B such that B A gives the matrix resulting from A after the given row operations are performed.

$$\begin{aligned} -3 R_{1} & \Rightarrow R_{1} \\ R_{1} & \Leftrightarrow R_{2} \\ 4 R_{1}+R_{2} & \Rightarrow R_{2} \end{aligned}$$

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. $\left(\frac{27}{64}\right)^{-4 / 3}$

Show how you would use a Strecker synthesis to make (a) leucine (b) valine (c) aspartic acid

Determine the linear transformation $T: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}$ that takes a vector of brand amounts (in hundreds of pounds) as input and gives the nutrient vector as output. Then find a formula for $T^{-1}$, and use it to determine the amount of Vigoro, Parker’s, and Bleyer’s required to produce the specified nutrient mix. 1092 pounds of nitrogen, 589 pounds of phosphoric acid, and 223 pounds of potash.