Consider the function $f(x)=\frac{e^{x}}{x-1}$. Find the equation of the tangent at the point where x = 2.

Permutation of $x_1,...,x_n$: ordering of $x_1,...,x_n$

Determine whether the given vectors $\mathbf{v}_{1}=(1,0,3 ,0), \mathbf{v}_{2}=(0,2,0,4), \mathbf{v}_{3}=(1,2,3,4)$ are linearly independent or linearly dependent. Do this essentially by inspection-that is, without solving a linear system of equations

The following account balances appear in the 2018 adjusted trial balance of Spiders Corporation: Common Stock, $30,000; Retained Earnings,$8,000; Dividends, $1,000; Service Revenue,$28,000; Salaries Expense, $16,000; and Rent Expense,$9,000. No common stock was issued during the year. Prepare the statement of stockholders' equity for the year ended December 31, 2018.

Find the reduced row echelon form R of the matrix A, and then find a matrix B for which BA = R.

$$A=\left[\begin{array}{lll}{1} & {0} & {0} \\ {2} & {4} & {0} \\ {4} & {8} & {0}\end{array}\right]$$

.

Solve for x exactly without using a calculator. $2 \ln (x-1)=\operatorname{In}\left(x^{2}-5\right)$

(a) Find a unit vector in the direction from

$$\left[ \begin{array} { r } { 3 } \\ { - 1 } \\ { 4 } \end{array} \right]$$

to

$$\left[ \begin{array} { l } { 1 } \\ { 3 } \\ { 5 } \end{array} \right]$$

. (b) If

$$\mathbf { u } \neq \mathbf { 0 }$$

, for which values of a is au a unit vector?

Let $A$ denote the $k \times k$ matrix

$$\left( \begin{array}{ccccc}{0} & {0} & {\cdots} & {0} & {-a_{0}} \\ {1} & {0} & {\cdots} & {0} & {-a_{1}} \\ {0} & {1} & {\cdots} & {0} & {-a_{2}} \\ {\vdots} & {\vdots} & { } & {\vdots} & {\vdots} \\ {0} & {0} & {\cdots} & {0} & {-a_{k-2}} \\ {0} & {0} & {\cdots} & {1} & {-a_{k-1}}\end{array}\right)$$

where $a_{0}, a_{1}, \ldots, a_{k-1}$ are arbitrary scalars. Prove that the characteristic polynomial of $A$ is $(-1)^{k}\left(a_{0}+a_{1} t+\cdots+a_{k-1} t^{k-1}+t^{k}\right)$

The normal distribution in probability is given by $p(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}}$ where $\sigma$ is the standard deviation and $\mu$ is the average. The standard normal distribution in probability, $p_{s}$ corresponds to $\mu=0\ \text { and } \ \sigma=1$. Compute the left endpoint estimates

$$R_{10}\ \text { and }\ R_{100}\ \text { of }\ \int_{-1}^{1} \frac{1}{\sqrt{2 \pi}} e^{-x^{2 }/2} d x$$

Find the angle of elevation of the top of a tree 13 m high from a point 25 m away on level ground.

Information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial function with real coefficients that has the zeros. Degree 4; zeros: $1,2,1+i$

Rocky and Arturo are designing a rocket for a competition. The top must be cone-shaped and the body of the rocket must be cylindrical. The volume of a cone is the product of

$$\frac { 1 } { 3 } , \pi$$

, the height h, and the square of the radius r. The volume of a cylinder is the product of

$$\pi$$

, the height t, and the square of the radius r. If the height of the body of the rocket is 9 inches, the height of the top is 5 inches, and the radius is 4 inches, find the volume of the rocket.

The ballad of "Sir Patrick Spens" may have a basis in historical fact. Do some research to find out whether or not such a person existed and what historical voyage the ballad may indirectly commemorate.

How does self-confidence affect people's ability to withstand stress?