Use a software program or a graphing utility to solve the system of linear equations.

$$\begin{align*} 3x+3y+12z &= 6\\ x+y+4z &= 2 \\ 2x+5y +20z &= 10 \\ -x +2y+8z &= 4 \end{align*}$$

The Michaelis-Menten equation models the rate v of an enzymatic reaction as a function of the concentration [S] of a substrate S. In the case of the enzyme chymotrypsin the equation is $v=\frac{0.14[\mathrm{S}]}{0.015+[\mathrm{S}]}$ Use a graphing calculator or computer to graph v as a function of [S].

Solve the system of linear equations.

$$\begin{align*} 0.05x_1 - 0.03x_2 &= 0.21 \\ 0.07x_1+0.02x_2 &= 0.17 \\ \end{align*}$$

Sketch the image of the unit square [a square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1)] under the specified transformation. T is the shear represented by T(x, y) = (x + 2y, y).

Use the bond dissociation enthalpy data in the table to calculate $\Delta H^{\circ}$ for (a) the monofluorination and (b) the monoiodination of methane. What is the main reason the values are so different?

Find the matrix A of the quadratic form associated with the equation. Then find the equation of the quadric surface in the rotated x'y'z' -system. $x^{2}+y^{2}+z^{2}+2 x y-8=0$

Prove that the symmetric matrix is diagonalizable.

$$A=\left[\begin{array}{lll} {0} & {a} & {0} \\ {a} & {0} & {a} \\ {0} & {a} & {0} \end{array}\right]$$

An evergreen nursery usually sells a type of shrub after 5 years of growth and shaping. The growth rate during those 5 years is approximated by $\frac{d h}{d t}=\frac{17.6 t}{\sqrt{17.6 t^{2}+1}}$ where t is the time (in years) and h is the height (in inches). The seedlings are 6 inches tall when planted (t=0). (a) Find the height function. (b) How tall are the shrubs when they are sold?

Suppose that the population dynamics of a species obeys a modified version of the logistic differential equation having the following form: $\frac{d N}{d t}=r\left(1-\frac{N}{K}\right)^{2} N$ where $r \neq 0 \text { and } K>0$ Apply the local stability criterion to the equilibrium $\hat{N}=K.$ What do you think your answer means about the stability of this equilibrium?

Show that the matrix is not diagonalizable.

$$\left[\begin{array}{ll} {0} & {0} \\ {k} & {0} \end{array}\right], k \neq 0$$

Find the local and absolute extreme values of the function on the given interval. $f(x)=x \sqrt{1-x}, \quad[-1,1]$

Why is the copious production of $\Lambda^{0}$ and K particles very difficult to reconcile with their slow decay, without the concept of strangeness? How does strangeness provide a reconciliation?

Rationalize each denominator. Assume that all radicands represent positive real numbers. $\frac{5}{\sqrt{10}}$

Determine whether the matrix is orthogonally diagonalizable.

$$\left[\begin{array}{rrr} {3} & {2} & {-3} \\ {-2} & {-1} & {2} \\ {-3} & {2} & {3} \end{array}\right]$$