Determine to six decimal places the steady state vector corresponding to the given initial state vector. Also find the smallest integer k such that $\mathbf{x}_{k}=\mathbf{x}_{k+1}$ to 6 decimal places for all entries. (NOTE: Even if it is less computationally efficient, it may be easier to compute state vectors using powers of A instead of the recursive formula.)

$$A=\left[\begin{array}{rrrr} 0 & 1 & 0.2 & 0.5 \\ 0.2 & 0 & 0.3 & 0 \\ 0.5 & 0 & 0.4 & 0.5 \\ 0.3 & 0 & 0.1 & 0 \end{array}\right], \ \mathbf{x}_{0}=\left[\begin{array}{c} 0.1 \\ 0.2 \\ 0.3 \\ 0.4 \end{array}\right]$$

Simplify each rational expression. Assume all variable expressions represent positive real numbers. $\frac{10\left(4 x^{2}-9\right)^{2}-25 x\left(4 x^{2}-9\right)^{3}}{15\left(4 x^{2}-9\right)^{6}}$

Find the normal line, in standard form, to y = f(x) at the indicated point. $y=\sqrt{3} x^{4}-2 \sqrt{3} x^{2}, \text { at } x=-\sqrt{3}$

Laurie mows lawns to earn extra money. She knows that she can mow at most 30 lawns in one week. She determines that she profits $15 on each lawn she mows. Identify a reasonable domain and range for this situation and draw a graph.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-linear plot. $y=6 \times 2^{-0.9 x}$

Find the domain of each rational expression. $\frac{12}{x^{2}+5 x+6}$

Describe the physiological role and mechanisms of extrinsic regulation of GFR.

All of the following characterize the ANS except (a) two-neuron efferent chain, (b) presence of nerve cell bodies in the CNS, (c) presence of nerve cell bodies in the ganglia, (d) innervation of skeletal muscles.

Bisphenol A is made on a large scale by a condensation of phenol with acetone. Suggest an appropriate catalyst, and propose a mechanism for this reaction.

Suppose that $x = (x_1, x_2, x_3)$ and $y = (y_1, y_2, y_3)$ are vectors in $R^3$. Then the cross product of x and y is given by

$$\mathbf{x} \times \mathbf{y}=\left[\begin{array}{l} x_{2} y_{3}-x_{3} y_{2} \\ x_{3} y_{1}-x_{1} y_{3} \\ x_{1} y_{2}-x_{2} y_{1} \end{array}\right]$$

Now let u be a fixed vector in $R^n$, and define $T(\mathbf{x})=\mathbf{u} \times \mathbf{x}$. Show that T is a linear transformation.

Michael and his two friends have a total of $9 to spend on tacos and burritos for lunch. Aburrito x costs$2 and a taco y costs $1. Find two solutions of the equation 2x + y = 9 to find how much food they can buy with$9. Explain each solution.

Show how you would use the Gabriel–malonic ester synthesis to make histidine. What stereochemistry would you expect in your synthetic product?

The distribution of resistance for resistors of a certain type is known to be normal, with 10% of all resistors having a resistance exceeding 10.256 ohms and 5% having a resistance smaller than 9.671 ohms. What are the mean value and standard deviation of the resistance distribution?

Let $f: D \rightarrow \mathbb{R}$ be continuous at $c \in D$ and suppose that f (c) > 0. Prove that there exists an $\alpha>0$ and a neighborhood U of c such that $f(x)>\alpha$ for all $x \in U \cap D$.